3010:
8358:
7106:
3566:
2641:
2046:
1606:
7084:
47:
1647:
156:
3254:
6542:
6514:
6484:
6454:
5892:
5846:
5816:
5786:
5740:
3005:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)\\&=&((p\land \lnot q)\lor \lnot p)&\land &((p\land \lnot q)\lor q)\\&=&((p\lor \lnot p)\land (\lnot q\lor \lnot p))&\land &((p\lor q)\land (\lnot q\lor q))\\&=&(\lnot p\lor \lnot q)&\land &(p\lor q)\\&=&\lnot (p\land q)&\land &(p\lor q)\end{matrix}}}
7050:
7020:
6986:
6802:
6772:
6672:
6642:
5496:
5466:
204:
8752:
6288:
6258:
6228:
6194:
3561:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)&\lor &(\lnot p\land q)&=&p{\overline {q}}+{\overline {p}}q\\&=&(p\lor q)&\land &(\lnot p\lor \lnot q)&=&(p+q)({\overline {p}}+{\overline {q}})\\&=&(p\lor q)&\land &\lnot (p\land q)&=&(p+q)({\overline {pq}})\end{matrix}}}
7401:. The XOR operation preserves randomness, meaning that a random bit XORed with a non-random bit will result in a random bit. Multiple sources of potentially random data can be combined using XOR, and the unpredictability of the output is guaranteed to be at least as good as the best individual source.
4236:
4993:
in 1890 (the original date is not definitely known, but almost certainly it is written after 1685; and 1890 is the publishing time). While both
Huntington in 1904 and Leibniz in 1890 used the symbol as an algebraic operation. Furthermore, Huntington in 1904 used the symbol as inclusive disjunction
7453:
XOR is also used to detect an overflow in the result of a signed binary arithmetic operation. If the leftmost retained bit of the result is not the same as the infinite number of digits to the left, then that means overflow occurred. XORing those two bits will give a "1" if there is an overflow.
3646:
4359:
The symbol used for exclusive disjunction varies from one field of application to the next, and even depends on the properties being emphasized in a given context of discussion. In addition to the abbreviation "XOR", any of the following symbols may also be seen:
3131:
2570:
3217:
4294:. In English, the disjunctive word "or" is often understood exclusively, particularly when used with the particle "either". The English example below would normally be understood in conversation as implying that Mary is not both a singer and a poet.
2458:
4106:
7429:) and writing it to another drive. Under this method, if any one of the three hard drives are lost, the lost byte can be re-created by XORing bytes from the remaining drives. For instance, if the drive containing
3583:
5985:
7363:
On some computer architectures, it is more efficient to store a zero in a register by XOR-ing the register with itself (bits XOR-ed with themselves are always zero) than to load and store the value zero.
270:
342:
1626:
4566:
7334:
5124:
for exclusive disjunction (it seems that the mistake spreads widely), while neither in 1929 nor in other works did Łukasiewicz make such use. In fact, in 1949 Bocheński introduced a system of
7273:
7170:
3047:
5330:
5361:
4864:
969:
2495:
1396:
911:
824:
8543:
3142:
1036:
697:
2280:
1969:
4700:
4673:
4526:
3039:
2487:
2249:
1942:
1355:
995:
5172:
as exclusive disjunction has no relationship with the Polish "alternatywa rozłączna" of "exclusive or" and is an accident for which see the table on page 16 of the book in 1949.
671:
7524:
6281:
6158:
3854:
3751:
2015:
1683:
1097:
8572:
7043:
6950:
6507:
6251:
6108:
5839:
5809:
5631:
5489:
5430:
4276:
4098:
4069:
3964:
2386:
3795:
1178:
6332:
4499:
is almost fixedly used nowadays) and exclusive disjunction, and may also bring about confusions with its other uses, some classical and modern textbooks still keep such use.
3997:
1915:
943:
783:
731:
611:
8648:
6665:
6606:
6477:
6354:
5886:
5780:
5708:
5576:
1508:
1456:
1062:
8673:
8448:
6795:
6736:
6536:
6413:
5034:
2035:
1717:. With two inputs, XOR is true if and only if the inputs differ (one is true, one is false). With multiple inputs, XOR is true if and only if the number of true inputs is
1313:
8619:
8419:
6874:
6446:
6392:
5669:
5609:
5094:
850:
5299:
4957:
1482:
757:
8394:
7014:
6901:
6714:
6634:
6222:
6059:
5458:
5408:
2165:
2092:
1422:
645:
376:
6584:
582:
533:
507:
6764:
4979:
3933:
3901:
3881:
3676:
2330:
1995:
1287:
1235:
876:
8702:
4884:
4808:
4786:
4722:
2613:
2593:
2378:
1201:
1128:
191:
8739:
8514:
8477:
8348:
7561:
4932:
4908:
4497:
3700:
2633:
2354:
4347:
This behavior of
English "or" is also found in other languages. However, many languages have disjunctive constructions which are robustly exclusive such as French
8077:
Huntington, E. V. (1933). "New Sets of
Independent Postulates for the Algebra of Logic, With Special Reference to Whitehead and Russell's Principia Mathematica".
6978:
6928:
6186:
6136:
6086:
6032:
4457:
4431:
2306:
2225:
one is true and the other is false. For example, if two horses are racing, then one of the two will win the race, but not both of them. The exclusive disjunction
1261:
5732:
5537:
1151:
556:
5273:
5253:
5170:
5150:
5122:
5064:
5012:
4750:
4646:
4626:
4606:
4586:
4477:
4405:
4381:
4040:
4020:
2138:
2117:
1757:
4231:{\displaystyle {\begin{matrix}r=p\land q&\Leftrightarrow &r=p\cdot q{\pmod {2}}\\r=p\oplus q&\Leftrightarrow &r=p+q{\pmod {2}}\\\end{matrix}}}
4302:
However, disjunction can also be understood inclusively, even in combination with "either". For instance, the first example below shows that "either" can be
1633:
452:
2059:
E.g. row AB corresponds to the 2-circle, and row ABC to the 3-circle Venn diagram shown above. (As in the Venn diagrams, white is false, and red is true.)
7704:
4306:
used in combination with an outright statement that both disjuncts are true. The second example shows that the exclusive inference vanishes away under
4310:
contexts. If disjunction were understood as exclusive in this example, it would leave open the possibility that some people ate both rice and beans.
4479:
is a connective. Furthermore, Boole used it exclusively. Although such use does not show the relationship between inclusive disjunction (for which
7379:
systems. XOR is also heavily used in block ciphers such as AES (Rijndael) or
Serpent and in block cipher implementation (CBC, CFB, OFB or CTR).
1732:
that case. Some informal ways of describing XOR are "one or the other but not both", "either one or the other", and "A or B, but not A and B".
8302:
4337:
7859:(2 ed.). San Diego, New York, Boston, London, Toronto, Sydney and Tokyo: A Harcourt Science and Technology Company. p. 51.
5916:
445:
223:
283:
3641:{\displaystyle \lnot (p\nleftrightarrow q)\Leftrightarrow \lnot p\nleftrightarrow q\Leftrightarrow p\nleftrightarrow \lnot q.}
7737:
17:
8126:
7817:
Jennings quotes numerous authors saying that the word "or" has an exclusive sense. See
Chapter 3, "The First Myth of 'Or'":
1619:
111:
4535:
438:
83:
7220:
As noted above, since exclusive disjunction is identical to addition modulo 2, the bitwise exclusive disjunction of two
7295:
3126:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&\lnot ((p\land q)\lor (\lnot p\land \lnot q))\end{matrix}}}
7230:
7127:
8295:
8275:
7766:
7578:
130:
4729:
90:
8276:
Proofs of XOR properties and applications of XOR, CS103: Mathematical
Foundations of Computing, Stanford University
7398:
2221:
Exclusive disjunction essentially means 'either one, but not both nor none'. In other words, the statement is true
8154:
5304:
5335:
4788:
as non-equivalence literally which is possibly because it could be defined from negation and equivalence easily.
7946:
7613:
5201:
4329:
2575:
This representation of XOR may be found useful when constructing a circuit or network, because it has only one
2565:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\land \lnot q)\lor (\lnot p\land q)\end{matrix}}}
68:
4817:
3212:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land (\lnot p\lor \lnot q)\end{matrix}}}
97:
5225:
948:
64:
1368:
881:
796:
8780:
8528:
8288:
8176:
1609:
1008:
676:
8053:
4914:." Note that the Latin word "aut" means "exclusive or" and "vel" means "inclusive or", and that Peano use
2254:
1947:
5217:
5209:
4678:
4651:
4504:
3018:
2466:
2453:{\displaystyle {\begin{matrix}p\nleftrightarrow q&=&(p\lor q)\land \lnot (p\land q)\end{matrix}}}
2228:
1920:
1326:
974:
79:
5132:
of classical logic which is a compatible extension of the notation of Łukasiewicz in 1929, and in which
650:
8180:
7509:
6266:
6143:
5097:
4986:
4325:
3815:
3712:
2000:
1656:
1067:
8557:
7028:
6935:
6492:
6236:
6093:
5824:
5794:
5616:
5474:
5415:
4252:
4074:
4045:
3940:
7383:
6298:
4990:
4333:
3805:. This unfortunately prevents the combination of these two systems into larger structures, such as a
3756:
1156:
6311:
5910:
3970:
1891:
922:
762:
710:
587:
8633:
7929:
7793:
7608:
6837:
6650:
6591:
6462:
6339:
5853:
5747:
5675:
5543:
5205:
5193:
3706:) are very useful in logic systems, they fail a more generalizable structure in the following way:
1487:
1435:
1041:
8658:
8433:
6780:
6721:
6521:
6398:
5019:
4702:
has the meaning of exclusive disjunction since the article is titled as "On the
Algebra of Logic".
2020:
1292:
8652:
8623:
8604:
8404:
7951:(in Russian) (3 ed.). МОСКВА: ГОСУДАРСТВЕННОЕ ИЗДАТЕЛЬСТВО ФИЗИКа-МАТЕМАТИЧЕСКОЙ ЛИТЕРАТУРЫ.
6847:
6419:
6365:
5642:
5582:
5073:
4529:
3656:
1556:
829:
276:
216:
57:
5278:
4939:
1461:
736:
155:
8379:
7886:
7628:
6993:
6880:
6693:
6613:
6201:
6038:
5437:
5387:
4757:
2144:
2071:
1736:
1526:
1401:
624:
355:
8112:
7974:
7754:
6563:
1705:
561:
512:
486:
8547:
8493:
7970:
7598:
7121:
7061:
6743:
5902:
5037:
4964:
4725:
4303:
4279:
3906:
3886:
3866:
3661:
3230:
2315:
1980:
1714:
1266:
1214:
855:
617:
417:
8687:
7841:
The
Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning
7717:
7285:
It is an optional bit-flipper (the deciding input chooses whether to invert the data input).
5275:, which may be interpreted as their elementwise exclusive or, has variously been denoted as
4869:
4793:
4771:
4707:
2598:
2578:
2363:
1183:
1110:
170:
8775:
8724:
8499:
8462:
8423:
8333:
7583:
7574:
7289:
5506:
5374:
5232:
5185:
4917:
4893:
4482:
3685:
3234:
2618:
2339:
1718:
1584:
703:
348:
35:
104:
8:
8677:
8452:
8200:
8118:
7839:
7593:
7387:
7353:
7117:
6957:
6907:
6165:
6115:
6065:
6011:
5988:
4436:
4410:
3802:
3703:
3679:
3577:
3223:
2309:
2285:
1594:
1240:
1207:
479:
7718:
5714:
5519:
1133:
538:
8785:
8312:
8007:
7483:
7468:
7458:
7213:
7072:
6812:
5258:
5238:
5155:
5135:
5129:
5107:
5049:
4997:
4735:
4631:
4611:
4591:
4571:
4462:
4390:
4366:
4307:
4025:
4005:
3806:
3238:
2123:
2102:
1742:
1710:
1589:
468:
407:
8142:
8038:
7360:
to add the numbers, and a series of AND, OR and NOT gates to create the carry output.
5101:
8756:
8576:
8323:
8197:
8122:
8025:
Huntington, E. V. (1904). "Sets of
Independent Postulates for the Algebra of Logic".
7762:
7733:
7487:
7475:
7100:
5909:
The exclusive or does not distribute over any binary function (not even itself), but
5189:
2054:
1531:
382:
8357:
8011:
4336:
and do not arise in downward entailing contexts if their calculation depends on the
1724:
It gains the name "exclusive or" because the meaning of "or" is ambiguous when both
8398:
8034:
7997:
7989:
7725:
7679:
5987:(Conjunction and exclusive or form the multiplication and addition operations of a
4959:
was used by Izrail
Solomonovich Gradshtein (Израиль Соломонович Градштейн) in 1936.
4321:
Examples such as the above have motivated analyses of the exclusivity inference as
4291:
4242:
1860:
1834:
1803:
1769:
1551:
1428:
7959:. Translated by Boddington, T. Oxford, London, New York and Paris: Pergamon Press.
7461:; however this is regarded as more of a curiosity and not encouraged in practice.
6840:
or self-inverse function; applying it twice leaves the variable input unchanged.
8168:
7648:
7479:
7464:
7422:
from two (or more) hard drives by XORing the just mentioned bytes, resulting in (
7345:
6821:
5125:
5067:
3999:
and has the added benefit of the arsenal of algebraic analysis tools for fields.
1536:
7993:
7844:. Cambridge/London: Macmillan, Barclay, & Macmillan/George Bell. p. 17.
2646:
8580:
8369:
8172:
7376:
7337:
7105:
6836:
Exclusive or with one specified input, as a function of the other input, is an
4982:
4811:
4765:
3242:
2222:
1760:
1546:
789:
7874:(3 ed.). New York, Dordrecht, Heidelberg and London: Springer. p. 3.
7729:
7371:, XOR is sometimes used as a simple, self-inverse mixing function, such as in
8769:
8718:
8714:
8327:
8280:
7922:
Notations de logique mathématique. Introduction au formulaire de mathématique
7623:
7618:
5041:
4340:. However, some researchers have treated exclusivity as a bona fide semantic
3860:
1561:
7408:
3–6 for creating parity information. For example, RAID can "back up" bytes
8270:
7724:. Translated by Bird, O. Dordrecht, Holland: D. Reidel Publishing Company.
7603:
7394:
7372:
7368:
7225:
5152:
for exclusive disjunction appeared at the first time. Bocheński's usage of
4761:
4753:
4384:
4042:
with 1, one can interpret the logical "AND" operation as multiplication on
2050:
1650:
8627:
8594:
8247:
7789:
7386:, modeling the XOR function requires a second layer because XOR is not a
5998:
2333:
2065:
1541:
1001:
163:
8002:
7669:
8427:
8222:
7638:
7341:
7088:
4341:
4322:
4246:
2045:
1579:
197:
7457:
XOR can be used to swap two numeric variables in computers, using the
7176:
Exclusive disjunction is often used for bitwise operations. Examples:
8551:
8373:
8205:
7800:(Winter 2016 ed.). Metaphysics Research Lab, Stanford University
7674:
1319:
7224:-bit strings is identical to the standard vector of addition in the
7083:
3259:
46:
8681:
8598:
8522:
8489:
7643:
7478:, XOR-based drawing methods are often used to manage such items as
7357:
7069:
4985:
in 1938. Shannon borrowed the symbol as exclusive disjunction from
2357:
1103:
31:
7906:
Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band
3248:
In summary, we have, in mathematical and in engineering notation:
8456:
7633:
7588:
6541:
6513:
6483:
6453:
5891:
5845:
5815:
5785:
5739:
4756:
in 1847, during the 40 years after Boole, his followers, such as
3967:. This field can represent any logic obtainable with the system
1725:
1646:
7340:
an odd number of the variables are true), which is equal to the
7712:(in French). The Netherlands: F. G. Kroonder, Bussum, Pays-Bas.
7109:
3798:
7982:
Transactions of the American Institute of Electrical Engineers
5980:{\displaystyle C\land (A\oplus B)=(C\land A)\oplus (C\land B)}
4111:
2216:
8195:
7503:
7278:
In computer science, exclusive disjunction has several uses:
5991:
5197:
5181:
3936:
1872:
1846:
1823:
1789:
265:{\displaystyle {\overline {x}}\cdot y+x\cdot {\overline {y}}}
7957:
Direct and Converse Theorems: The Elements of Symbolic Logic
7467:
leverage XOR properties in order to save space to represent
337:{\displaystyle ({\overline {x}}+{\overline {y}})\cdot (x+y)}
7891:
Studies in Logic by Members of the Johns Hopkins University
7405:
7049:
7019:
6985:
6801:
6771:
6671:
6641:
5495:
5465:
5213:
1809:
1775:
203:
8223:"Exclusive OR (XOR) and hardware random number generators"
7526:). Apart from the ASCII codes, the operator is encoded at
4344:
and proposed nonclassical logics which would validate it.
6287:
6257:
6227:
6193:
5221:
3229:
The exclusive or is also equivalent to the negation of a
3237:
is equivalent to the disjunction of the negation of its
2094:
shows that it outputs true whenever the inputs differ:
7893:. Boston: Little, Brown & Company. pp. 17–71.
5994:, and as in any field they obey the distributive law.)
3147:
3052:
2500:
2391:
8727:
8690:
8661:
8636:
8607:
8560:
8531:
8502:
8465:
8436:
8407:
8382:
8336:
8100:. New Jersey: Princeton University Press. p. 37.
7975:"A Symbolic Analysis of Relay and Switching Circuits"
7908:(in German). Leipzig: Druck und Verlag B. G. Teubner.
7512:
7298:
7233:
7130:
7031:
6996:
6960:
6938:
6910:
6883:
6850:
6783:
6746:
6724:
6696:
6653:
6616:
6594:
6566:
6524:
6495:
6465:
6422:
6401:
6368:
6342:
6314:
6269:
6239:
6204:
6168:
6146:
6118:
6096:
6068:
6041:
6014:
5919:
5856:
5827:
5797:
5750:
5717:
5678:
5645:
5619:
5585:
5546:
5522:
5477:
5440:
5418:
5390:
5338:
5307:
5281:
5261:
5241:
5158:
5138:
5110:
5076:
5052:
5022:
5000:
4967:
4942:
4920:
4896:
4872:
4820:
4796:
4774:
4738:
4710:
4681:
4654:
4634:
4614:
4594:
4574:
4538:
4507:
4485:
4465:
4439:
4413:
4393:
4369:
4255:
4109:
4077:
4048:
4028:
4008:
3973:
3943:
3909:
3889:
3869:
3818:
3759:
3715:
3688:
3664:
3586:
3257:
3145:
3050:
3021:
2644:
2635:
operations. A proof of this identity is given below:
2621:
2601:
2581:
2498:
2469:
2389:
2366:
2342:
2318:
2288:
2257:
2231:
2147:
2126:
2105:
2074:
2023:
2003:
1983:
1950:
1923:
1894:
1869:
1866:
1843:
1840:
1815:
1812:
1806:
1781:
1778:
1772:
1745:
1659:
1490:
1464:
1438:
1404:
1371:
1329:
1295:
1269:
1243:
1217:
1186:
1159:
1136:
1113:
1070:
1044:
1011:
977:
951:
925:
884:
858:
832:
799:
765:
739:
713:
679:
653:
627:
590:
564:
541:
515:
489:
358:
286:
226:
173:
7124:
representation. This is also the vector addition in
4561:{\displaystyle A\operatorname {\overline {\vee }} B}
4407:
mainly on classes, he also considered the case that
1849:
1792:
6556:When all inputs are true, the output is not true.
4285:
1863:
1837:
1820:
1786:
71:. Unsourced material may be challenged and removed.
8733:
8696:
8667:
8642:
8613:
8566:
8537:
8508:
8471:
8442:
8413:
8388:
8342:
7948:ПРЯМАЯ И ОБРАТНАЯ ТЕОРЕМЫ: ЭЛЕМЕНТЫ АЛГЕБРЫ ЛОГИКИ
7755:"9.2: Algebraic normal forms of Boolean functions"
7518:
7328:
7267:
7164:
7037:
7008:
6972:
6944:
6922:
6895:
6868:
6789:
6758:
6730:
6708:
6659:
6628:
6600:
6578:
6530:
6501:
6471:
6440:
6407:
6386:
6348:
6326:
6275:
6245:
6216:
6180:
6152:
6130:
6102:
6080:
6053:
6026:
5979:
5880:
5833:
5803:
5774:
5726:
5702:
5663:
5625:
5603:
5570:
5531:
5483:
5452:
5424:
5402:
5355:
5324:
5293:
5267:
5247:
5164:
5144:
5116:
5088:
5058:
5028:
5006:
4973:
4951:
4926:
4902:
4878:
4858:
4802:
4780:
4744:
4716:
4694:
4667:
4640:
4620:
4600:
4580:
4560:
4520:
4491:
4471:
4451:
4425:
4399:
4375:
4270:
4230:
4092:
4063:
4034:
4014:
3991:
3958:
3927:
3895:
3875:
3848:
3789:
3745:
3694:
3670:
3640:
3560:
3211:
3125:
3033:
3004:
2627:
2607:
2587:
2564:
2481:
2452:
2372:
2348:
2324:
2300:
2274:
2243:
2159:
2132:
2111:
2086:
2029:
2009:
1989:
1963:
1936:
1909:
1751:
1677:
1502:
1476:
1450:
1416:
1390:
1349:
1307:
1281:
1255:
1229:
1195:
1172:
1145:
1122:
1091:
1056:
1030:
989:
963:
937:
905:
870:
844:
818:
777:
751:
725:
691:
665:
639:
605:
576:
550:
527:
501:
370:
336:
264:
185:
8079:Transactions of the American Mathematical Society
8027:Transactions of the American Mathematical Society
7329:{\displaystyle A\oplus B\oplus C\oplus D\oplus E}
6686:When all inputs are false, the output is false.
5911:logical conjunction distributes over exclusive or
8767:
8167:
8153:] (in Polish) (1 ed.). Warsaw, Poland:
7667:
7268:{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{n}}
7165:{\displaystyle (\mathbb {Z} /2\mathbb {Z} )^{4}}
3222:This equivalence can be established by applying
8135:
8052:Leibniz, G. W. (1890) . Gerhardt, C. I. (ed.).
7698:
7696:
4290:Disjunction is often understood exclusively in
8310:
8114:Routledge Encyclopedia of Philosophy, Volume 8
8070:
8018:
7938:
7087:Traditional symbolic representation of an XOR
30:"XOR" redirects here. For the logic gate, see
8296:
7934:. Roma: Edizioni Cremonese. pp. 123–176.
7863:
6810:
4989:in 1904. Huntington borrowed the symbol from
4314:2. Mary is either a singer or a poet or both.
4278:, using this basis, is called the function's
3650:
3226:twice to the fourth line of the above proof.
1627:
446:
27:True when either but not both inputs are true
7897:
7872:A Concise Introduction to Mathematical Logic
7693:
6680:
3922:
3910:
3834:
3822:
3775:
3763:
3731:
3719:
2489:can also be expressed in the following way:
8141:
8055:Die philosophischen Schriften, Siebter Band
7848:
7833:
7831:
5325:{\displaystyle S\mathop {\triangledown } T}
3571:
2217:Equivalences, elimination, and introduction
8303:
8289:
8104:
8089:
8076:
8058:(in German). Berlin: Weidmann. p. 237
8024:
7954:
7869:
7784:
7782:
7780:
7778:
7498:It is also called "not left-right arrow" (
5356:{\displaystyle S\mathop {\vartriangle } T}
3233:, by the rules of material implication (a
1634:
1620:
453:
439:
8001:
7944:
7913:
7715:
7702:
7393:Similarly, XOR can be used in generating
7251:
7238:
7148:
7135:
5100:in 1949. Somebody may mistake that it is
4846:
4842:
4258:
4080:
4051:
3946:
131:Learn how and when to remove this message
7903:
7854:
7828:
7819:
7212:(this is equivalent to addition without
7104:
7082:
7064:values for true (1) and false (0), then
4859:{\displaystyle a\circ b=a-b\,\cup \,b-a}
4328:calculated on the basis of an inclusive
2057:XOR of the arguments shown on the left.
2044:
1645:
8051:
7969:
7878:
7798:The Stanford Encyclopedia of Philosophy
7775:
7450:can be XORed to recover the lost byte.
6550:
4071:and the "XOR" operation as addition on
3812:However, the system using exclusive or
964:{\displaystyle A\not \Leftrightarrow B}
14:
8768:
8220:
8095:
7661:
7282:It tells whether two bits are unequal.
5192:exclusive or operator, beginning with
4752:as equivalence could be dated back to
4532:in 1883. Strictly speaking, Ladd used
4354:
1391:{\displaystyle A{\underline {\lor }}B}
906:{\displaystyle {\overline {A\cdot B}}}
819:{\displaystyle A{\overline {\land }}B}
8538:{\displaystyle \not \leftrightarrow }
8284:
8245:
8221:Davies, Robert B (28 February 2002).
8196:
8110:
7927:
7919:
7837:
7788:
7668:Germundsson, Roger; Weisstein, Eric.
5504:
5372:
4002:More specifically, if one associates
1031:{\displaystyle A{\overline {\lor }}B}
692:{\displaystyle A\leftrightharpoons B}
7910:Reprinted by Thoemmes Press in 2000.
7884:
7857:A Mathematical Introduction to Logic
7824:. New York: Oxford University Press.
7752:
7094:
4994:(logical sum) too, and in 1933 used
2275:{\displaystyle p\operatorname {?} q}
1964:{\displaystyle {\underline {\vee }}}
69:adding citations to reliable sources
40:
7078:
6818:
6296:
4695:{\displaystyle {\overline {\vee }}}
4668:{\displaystyle {\overline {\vee }}}
4521:{\displaystyle {\overline {\vee }}}
4317:3. Nobody ate either rice or beans.
4216:
4209:
4159:
4152:
3034:{\displaystyle p\nleftrightarrow q}
2482:{\displaystyle p\nleftrightarrow q}
2308:, can be expressed in terms of the
2244:{\displaystyle p\nleftrightarrow q}
1937:{\displaystyle {\overline {\vee }}}
1350:{\displaystyle A\ {\text{XNOR}}\ B}
990:{\displaystyle A\nleftrightarrow B}
24:
8728:
8503:
8337:
8098:Introduction to Mathematical Logic
5996:
5900:
3629:
3608:
3587:
3493:
3401:
3392:
3306:
3287:
3196:
3187:
3107:
3098:
3071:
2958:
2917:
2908:
2877:
2834:
2825:
2810:
2770:
2742:
2730:
2693:
2674:
2582:
2543:
2528:
2428:
2367:
1114:
666:{\displaystyle A\Leftrightarrow B}
597:
594:
568:
25:
8797:
8264:
8039:10.1090/S0002-9947-1904-1500675-4
7519:{\displaystyle \nleftrightarrow }
7399:hardware random number generators
6276:{\displaystyle \nLeftrightarrow }
6153:{\displaystyle \nLeftrightarrow }
3849:{\displaystyle (\{T,F\},\oplus )}
3746:{\displaystyle (\{T,F\},\wedge )}
2010:{\displaystyle \nleftrightarrow }
1678:{\displaystyle A\oplus B\oplus C}
1092:{\displaystyle {\overline {A+B}}}
8750:
8567:{\displaystyle \leftrightarrow }
8356:
8185:. Prentice-Hall. pp. 44–46.
8177:"2.9: Bitwise logical operators"
7048:
7038:{\displaystyle \Leftrightarrow }
7018:
6984:
6945:{\displaystyle \Leftrightarrow }
6800:
6770:
6670:
6640:
6540:
6512:
6502:{\displaystyle \Leftrightarrow }
6482:
6452:
6286:
6256:
6246:{\displaystyle \Leftrightarrow }
6226:
6192:
6103:{\displaystyle \Leftrightarrow }
5890:
5844:
5834:{\displaystyle \Leftrightarrow }
5814:
5804:{\displaystyle \Leftrightarrow }
5784:
5738:
5626:{\displaystyle \Leftrightarrow }
5494:
5484:{\displaystyle \Leftrightarrow }
5464:
5425:{\displaystyle \Leftrightarrow }
5036:, also denoting the negation of
4675:as exclusions, while implicitly
4286:Exclusive or in natural language
4271:{\displaystyle \mathbb {F} _{2}}
4093:{\displaystyle \mathbb {F} _{2}}
4064:{\displaystyle \mathbb {F} _{2}}
3959:{\displaystyle \mathbb {F} _{2}}
3863:. The combination of operators
3015:It is sometimes useful to write
1859:
1833:
1802:
1768:
1605:
1604:
202:
154:
45:
8246:Nobel, Rickard (26 July 2011).
8239:
8214:
8189:
8161:
8045:
7963:
7761:. CRC Press. pp. 285–286.
4732:in 1890, Although the usage of
3790:{\displaystyle (\{T,F\},\lor )}
1173:{\displaystyle {\overline {A}}}
56:needs additional citations for
8608:
8561:
8437:
8408:
8383:
8151:Elements of Mathematical Logic
7811:
7746:
7720:A Precis of Mathematical Logic
7706:Précis de logique mathématique
7614:List of Boolean algebra topics
7382:In simple threshold-activated
7352:In logical circuits, a simple
7256:
7234:
7153:
7131:
7032:
6939:
6830:
6784:
6725:
6525:
6496:
6435:
6423:
6402:
6381:
6369:
6327:{\displaystyle A\rightarrow B}
6318:
6240:
6097:
5974:
5962:
5956:
5944:
5938:
5926:
5828:
5798:
5658:
5646:
5620:
5598:
5586:
5478:
5419:
4298:1. Mary is a singer or a poet.
4220:
4210:
4188:
4163:
4153:
4131:
3992:{\displaystyle (\land ,\lor )}
3986:
3974:
3843:
3819:
3784:
3760:
3740:
3716:
3620:
3605:
3602:
3590:
3551:
3533:
3530:
3518:
3508:
3496:
3483:
3471:
3458:
3432:
3429:
3417:
3407:
3389:
3379:
3367:
3318:
3303:
3293:
3278:
3202:
3184:
3178:
3166:
3116:
3113:
3095:
3089:
3077:
3074:
2995:
2983:
2973:
2961:
2945:
2933:
2923:
2905:
2892:
2889:
2874:
2868:
2856:
2853:
2843:
2840:
2822:
2816:
2801:
2798:
2785:
2776:
2761:
2758:
2748:
2736:
2721:
2718:
2705:
2690:
2680:
2665:
2595:operation and small number of
2555:
2540:
2534:
2519:
2443:
2431:
2422:
2410:
1910:{\displaystyle {\dot {\vee }}}
1494:
1442:
1048:
938:{\displaystyle A\not \equiv B}
836:
778:{\displaystyle A\rightarrow B}
769:
726:{\displaystyle A\Rightarrow B}
717:
683:
657:
606:{\displaystyle A\&\&B}
331:
319:
313:
287:
180:
174:
13:
1:
8643:{\displaystyle \nrightarrow }
8155:Państwowe Wydawnictwo Naukowe
8147:Elementy logiki matematycznej
7796:. In Zalta, Edward N. (ed.).
7558:⊕, ⊕
7288:It tells whether there is an
6660:{\displaystyle \nRightarrow }
6601:{\displaystyle \nRightarrow }
6472:{\displaystyle \nRightarrow }
6349:{\displaystyle \nRightarrow }
5881:{\displaystyle ~~~\oplus ~~~}
5775:{\displaystyle ~~~\oplus ~~~}
5703:{\displaystyle ~~~\oplus ~~~}
5571:{\displaystyle ~~~\oplus ~~~}
5367:
4387:in 1847. Although Boole used
4332:. Implicatures are typically
2040:
1503:{\displaystyle A\leftarrow B}
1451:{\displaystyle A\Leftarrow B}
1057:{\displaystyle A\downarrow B}
8668:{\displaystyle \nleftarrow }
8443:{\displaystyle \rightarrow }
7822:The Genealogy of Disjunction
7493:
6790:{\displaystyle \Rightarrow }
6731:{\displaystyle \Rightarrow }
6531:{\displaystyle \rightarrow }
6408:{\displaystyle \rightarrow }
5029:{\displaystyle \not \equiv }
4687:
4660:
4547:
4513:
3546:
3453:
3440:
3349:
3336:
2030:{\displaystyle \not \equiv }
1929:
1308:{\displaystyle A\parallel B}
1165:
1084:
1020:
898:
808:
308:
295:
257:
232:
7:
8614:{\displaystyle \downarrow }
8414:{\displaystyle \leftarrow }
8248:"How RAID 5 actually works"
7994:10.1109/T-AIEE.1938.5057767
7567:
6869:{\displaystyle ~A\oplus B~}
6441:{\displaystyle (B\oplus C)}
6387:{\displaystyle (A\oplus C)}
5664:{\displaystyle (A\oplus B)}
5604:{\displaystyle (B\oplus C)}
5184:, has been used in several
5089:{\displaystyle J\phi \psi }
4724:, denoting the negation of
4326:conversational implicatures
845:{\displaystyle A\uparrow B}
10:
8802:
8182:The C Programming Language
7955:Gradshtein, I. S. (1963).
7945:ГРАДШТЕЙН, И. С. (1959) .
7889:. In Peirce, C. S. (ed.).
7384:artificial neural networks
7098:
5294:{\displaystyle S\ominus T}
4987:Edward Vermilye Huntington
4952:{\displaystyle \vee \vee }
3651:Relation to modern algebra
3576:By applying the spirit of
2463:The exclusive disjunction
2053:is the truth table of the
1477:{\displaystyle A\subset B}
752:{\displaystyle A\supset B}
29:
8747:
8710:
8590:
8485:
8389:{\displaystyle \uparrow }
8365:
8354:
8319:
7924:. Turin: Fratelli Boccna.
7887:"On the Algebra of Logic"
7759:Algorithmic Cryptanalysis
7730:10.1007/978-94-017-0592-9
7716:Bocheński, J. M. (1959).
7703:Bocheński, J. M. (1949).
7009:{\displaystyle ~\oplus ~}
6896:{\displaystyle ~\oplus ~}
6709:{\displaystyle A\oplus B}
6682:Falsehood-preserving: yes
6629:{\displaystyle A\oplus B}
6217:{\displaystyle ~\oplus ~}
6054:{\displaystyle ~\oplus ~}
5453:{\displaystyle B\oplus A}
5403:{\displaystyle A\oplus B}
5128:that names all 16 binary
5014:as inclusive disjunction.
4991:Gottfried Wilhelm Leibniz
4934:as inclusive disjunction.
3241:and its consequence) and
2160:{\displaystyle A\oplus B}
2087:{\displaystyle A\oplus B}
1417:{\displaystyle A\oplus B}
640:{\displaystyle A\equiv B}
434:
426:
416:
406:
398:
390:
381:
371:{\displaystyle x\oplus y}
347:
275:
215:
210:
196:
162:
153:
148:
7885:Ladd, Christine (1883).
7870:Rautenberg, W. (2010) .
7820:Jennings, R. E. (1994).
7655:
7045:
7025:
6952:
6932:
6797:
6777:
6738:
6718:
6667:
6647:
6608:
6588:
6579:{\displaystyle A\land B}
6509:
6489:
6479:
6459:
6356:
6336:
6283:
6263:
6253:
6233:
6160:
6140:
6110:
6090:
5841:
5821:
5811:
5791:
5633:
5613:
5491:
5471:
5432:
5412:
5104:who is the first to use
3572:Negation of the operator
2049:Each row of this binary
577:{\displaystyle A\&B}
528:{\displaystyle A\cdot B}
502:{\displaystyle A\land B}
8653:Converse nonimplication
7931:Opere Scelte, Volume II
7292:number of 1 bits (
6828:The function is linear.
6759:{\displaystyle A\lor B}
4974:{\displaystyle \oplus }
4768:and so on, did not use
4530:Christine Ladd-Franklin
3935:produce the well-known
3928:{\displaystyle \{T,F\}}
3896:{\displaystyle \oplus }
3876:{\displaystyle \wedge }
3671:{\displaystyle \wedge }
2325:{\displaystyle \wedge }
1990:{\displaystyle \oplus }
1739:by the prefix operator
1701:logical non-equivalence
1557:Functional completeness
1282:{\displaystyle A\mid B}
1230:{\displaystyle A\lor B}
871:{\displaystyle A\mid B}
8735:
8698:
8697:{\displaystyle \land }
8669:
8644:
8615:
8568:
8539:
8510:
8473:
8444:
8415:
8390:
8344:
8201:"Symmetric Difference"
8111:Craig, Edward (1998).
7855:Enderton, H. (2001) .
7753:Joux, Antoine (2009).
7629:Propositional calculus
7562:mathematical operators
7520:
7330:
7269:
7173:
7166:
7091:
7039:
7010:
6974:
6946:
6924:
6897:
6870:
6791:
6760:
6732:
6710:
6661:
6630:
6602:
6580:
6532:
6503:
6473:
6442:
6409:
6388:
6350:
6328:
6277:
6247:
6218:
6182:
6154:
6132:
6104:
6082:
6055:
6028:
5981:
5882:
5835:
5805:
5776:
5728:
5704:
5665:
5627:
5605:
5572:
5533:
5485:
5454:
5426:
5404:
5357:
5326:
5295:
5269:
5249:
5166:
5146:
5118:
5090:
5060:
5030:
5008:
4975:
4953:
4928:
4904:
4880:
4879:{\displaystyle \circ }
4860:
4804:
4803:{\displaystyle \circ }
4782:
4781:{\displaystyle \not =}
4758:Charles Sanders Peirce
4746:
4718:
4717:{\displaystyle \not =}
4696:
4669:
4642:
4622:
4602:
4582:
4562:
4522:
4493:
4473:
4453:
4427:
4401:
4377:
4272:
4232:
4094:
4065:
4036:
4016:
3993:
3960:
3929:
3897:
3877:
3850:
3791:
3747:
3696:
3672:
3642:
3562:
3213:
3127:
3041:in the following way:
3035:
3006:
2629:
2609:
2608:{\displaystyle \land }
2589:
2588:{\displaystyle \lnot }
2566:
2483:
2454:
2374:
2373:{\displaystyle \lnot }
2350:
2326:
2302:
2276:
2245:
2161:
2134:
2113:
2088:
2061:
2031:
2011:
1991:
1965:
1938:
1911:
1753:
1713:whose negation is the
1685:
1679:
1527:Propositional calculus
1504:
1478:
1452:
1418:
1392:
1351:
1309:
1283:
1257:
1231:
1197:
1196:{\displaystyle \sim A}
1174:
1147:
1124:
1123:{\displaystyle \neg A}
1093:
1058:
1032:
991:
965:
939:
907:
872:
846:
820:
779:
753:
727:
693:
667:
641:
607:
578:
552:
529:
503:
372:
338:
266:
187:
186:{\displaystyle (0110)}
34:. For other uses, see
8757:Philosophy portal
8736:
8734:{\displaystyle \bot }
8699:
8670:
8645:
8616:
8569:
8540:
8511:
8509:{\displaystyle \neg }
8474:
8472:{\displaystyle \lor }
8445:
8416:
8391:
8345:
8343:{\displaystyle \top }
7904:Schröder, E. (1890).
7599:Disjunctive syllogism
7521:
7331:
7270:
7167:
7108:
7086:
7040:
7011:
6975:
6947:
6925:
6898:
6871:
6792:
6761:
6733:
6711:
6662:
6631:
6603:
6581:
6533:
6504:
6474:
6443:
6410:
6389:
6351:
6329:
6278:
6248:
6219:
6183:
6155:
6133:
6105:
6083:
6056:
6029:
5982:
5883:
5836:
5806:
5777:
5729:
5705:
5666:
5628:
5606:
5573:
5534:
5486:
5455:
5427:
5405:
5358:
5327:
5296:
5270:
5250:
5186:programming languages
5167:
5147:
5119:
5098:Józef Maria Bocheński
5091:
5061:
5031:
5009:
4976:
4954:
4929:
4927:{\displaystyle \cup }
4905:
4903:{\displaystyle \cup }
4886:corresponds to Latin
4881:
4861:
4805:
4783:
4747:
4719:
4697:
4670:
4643:
4623:
4603:
4583:
4563:
4523:
4494:
4492:{\displaystyle \vee }
4474:
4454:
4428:
4402:
4378:
4280:algebraic normal form
4273:
4241:The description of a
4233:
4095:
4066:
4037:
4017:
3994:
3961:
3930:
3898:
3878:
3851:
3792:
3748:
3697:
3695:{\displaystyle \lor }
3673:
3643:
3563:
3231:logical biconditional
3214:
3128:
3036:
3007:
2630:
2628:{\displaystyle \lor }
2610:
2590:
2567:
2484:
2455:
2375:
2351:
2349:{\displaystyle \lor }
2327:
2303:
2277:
2246:
2162:
2135:
2114:
2089:
2048:
2032:
2012:
1992:
1966:
1939:
1912:
1754:
1715:logical biconditional
1697:exclusive alternation
1693:exclusive disjunction
1680:
1649:
1585:Programming languages
1505:
1479:
1453:
1419:
1393:
1352:
1310:
1284:
1258:
1232:
1198:
1175:
1148:
1125:
1094:
1059:
1033:
992:
966:
940:
908:
873:
847:
821:
780:
754:
728:
694:
668:
642:
608:
579:
553:
530:
504:
373:
339:
267:
188:
18:Exclusive disjunction
8725:
8688:
8659:
8634:
8605:
8558:
8529:
8500:
8463:
8434:
8405:
8399:Converse implication
8380:
8334:
8119:Taylor & Francis
8096:Church, A. (1996) .
7584:Affirming a disjunct
7575:Material conditional
7510:
7356:can be made with an
7296:
7231:
7128:
7118:nonnegative integers
7029:
6994:
6958:
6936:
6908:
6881:
6848:
6781:
6744:
6722:
6694:
6651:
6614:
6592:
6564:
6552:Truth-preserving: no
6522:
6493:
6463:
6420:
6399:
6366:
6340:
6312:
6267:
6237:
6202:
6166:
6144:
6116:
6094:
6066:
6039:
6012:
5917:
5854:
5825:
5795:
5748:
5715:
5676:
5643:
5617:
5583:
5544:
5520:
5475:
5438:
5416:
5388:
5336:
5305:
5279:
5259:
5239:
5233:symmetric difference
5156:
5136:
5108:
5074:
5050:
5020:
4998:
4965:
4940:
4918:
4894:
4870:
4818:
4794:
4772:
4736:
4708:
4679:
4652:
4632:
4612:
4592:
4572:
4536:
4505:
4483:
4463:
4437:
4433:are propositions in
4411:
4391:
4367:
4253:
4107:
4075:
4046:
4026:
4006:
3971:
3941:
3907:
3887:
3867:
3816:
3757:
3713:
3686:
3662:
3584:
3255:
3243:material equivalence
3235:material conditional
3143:
3048:
3019:
2642:
2619:
2599:
2579:
2496:
2467:
2387:
2364:
2340:
2316:
2286:
2255:
2229:
2145:
2124:
2103:
2072:
2021:
2001:
1981:
1948:
1921:
1892:
1743:
1657:
1488:
1462:
1436:
1402:
1369:
1327:
1293:
1267:
1241:
1215:
1184:
1157:
1134:
1111:
1068:
1042:
1009:
975:
949:
923:
882:
856:
830:
797:
763:
737:
711:
677:
651:
625:
588:
562:
539:
513:
487:
356:
349:Zhegalkin polynomial
284:
224:
171:
65:improve this article
36:XOR (disambiguation)
8781:Logical connectives
8313:logical connectives
8169:Kernighan, Brian W.
7594:Controlled NOT gate
7490:or overlay planes.
7486:on systems without
7068:works exactly like
6973:{\displaystyle ~A~}
6923:{\displaystyle ~B~}
6181:{\displaystyle ~A~}
6131:{\displaystyle ~0~}
6081:{\displaystyle ~A~}
6027:{\displaystyle ~A~}
5196:and also including
4452:{\displaystyle x+y}
4426:{\displaystyle x,y}
4355:Alternative symbols
3801:, but neither is a
2310:logical conjunction
2301:{\displaystyle Jpq}
1595:Philosophy of logic
1256:{\displaystyle A+B}
469:Logical connectives
145:
8731:
8694:
8665:
8640:
8611:
8564:
8535:
8506:
8469:
8440:
8411:
8386:
8370:Alternative denial
8340:
8198:Weisstein, Eric W.
8173:Ritchie, Dennis M.
7928:Peano, G. (1958).
7920:Peano, G. (1894).
7838:Boole, G. (1847).
7516:
7469:doubly linked list
7459:XOR swap algorithm
7388:linearly separable
7326:
7265:
7174:
7162:
7092:
7035:
7006:
6970:
6942:
6920:
6893:
6866:
6787:
6756:
6728:
6706:
6657:
6626:
6598:
6576:
6528:
6499:
6469:
6438:
6405:
6384:
6346:
6324:
6273:
6243:
6214:
6178:
6150:
6128:
6100:
6078:
6051:
6024:
5977:
5878:
5831:
5801:
5772:
5727:{\displaystyle ~C}
5724:
5700:
5661:
5623:
5601:
5568:
5532:{\displaystyle ~A}
5529:
5481:
5450:
5422:
5400:
5353:
5322:
5291:
5265:
5245:
5162:
5142:
5114:
5086:
5056:
5026:
5004:
4971:
4949:
4924:
4900:
4876:
4856:
4800:
4778:
4742:
4714:
4692:
4665:
4638:
4618:
4598:
4578:
4558:
4518:
4489:
4469:
4459:, and at the time
4449:
4423:
4397:
4373:
4308:downward entailing
4268:
4228:
4226:
4090:
4061:
4032:
4012:
3989:
3956:
3937:two-element field
3925:
3893:
3873:
3846:
3787:
3743:
3692:
3668:
3638:
3558:
3556:
3209:
3207:
3123:
3121:
3031:
3002:
3000:
2625:
2605:
2585:
2562:
2560:
2479:
2450:
2448:
2370:
2346:
2322:
2298:
2272:
2251:, also denoted by
2241:
2157:
2130:
2109:
2084:
2062:
2027:
2007:
1987:
1961:
1959:
1934:
1907:
1749:
1706:logical inequality
1686:
1675:
1590:Mathematical logic
1500:
1474:
1448:
1414:
1388:
1383:
1347:
1305:
1279:
1253:
1227:
1193:
1170:
1146:{\displaystyle -A}
1143:
1120:
1089:
1054:
1028:
987:
961:
935:
903:
868:
842:
816:
775:
749:
723:
689:
663:
637:
603:
574:
551:{\displaystyle AB}
548:
525:
499:
368:
334:
262:
183:
143:
8763:
8762:
7739:978-90-481-8329-6
7560:), both in block
7506:-based markdown (
7476:computer graphics
7471:data structures.
7101:Bitwise operation
7095:Bitwise operation
7056:
7055:
7005:
6999:
6969:
6963:
6919:
6913:
6892:
6886:
6865:
6853:
6808:
6807:
6678:
6677:
6548:
6547:
6294:
6293:
6213:
6207:
6177:
6171:
6127:
6121:
6077:
6071:
6050:
6044:
6023:
6017:
5898:
5897:
5877:
5874:
5871:
5865:
5862:
5859:
5771:
5768:
5765:
5759:
5756:
5753:
5720:
5699:
5696:
5693:
5687:
5684:
5681:
5567:
5564:
5561:
5555:
5552:
5549:
5525:
5502:
5501:
5268:{\displaystyle T}
5248:{\displaystyle S}
5165:{\displaystyle J}
5145:{\displaystyle J}
5117:{\displaystyle J}
5059:{\displaystyle J}
5007:{\displaystyle +}
4745:{\displaystyle =}
4690:
4663:
4641:{\displaystyle B}
4621:{\displaystyle A}
4601:{\displaystyle B}
4581:{\displaystyle A}
4550:
4516:
4472:{\displaystyle +}
4400:{\displaystyle +}
4376:{\displaystyle +}
4338:Maxim of Quantity
4292:natural languages
4035:{\displaystyle T}
4015:{\displaystyle F}
3807:mathematical ring
3549:
3456:
3443:
3352:
3339:
2214:
2213:
2133:{\displaystyle B}
2112:{\displaystyle A}
2060:
1952:
1932:
1904:
1752:{\displaystyle J}
1644:
1643:
1513:
1512:
1376:
1343:
1339:
1335:
1168:
1087:
1023:
901:
811:
463:
462:
311:
298:
260:
235:
141:
140:
133:
115:
16:(Redirected from
8793:
8755:
8754:
8753:
8740:
8738:
8737:
8732:
8703:
8701:
8700:
8695:
8674:
8672:
8671:
8666:
8649:
8647:
8646:
8641:
8620:
8618:
8617:
8612:
8573:
8571:
8570:
8565:
8544:
8542:
8541:
8536:
8515:
8513:
8512:
8507:
8478:
8476:
8475:
8470:
8449:
8447:
8446:
8441:
8420:
8418:
8417:
8412:
8395:
8393:
8392:
8387:
8360:
8349:
8347:
8346:
8341:
8305:
8298:
8291:
8282:
8281:
8259:
8258:
8256:
8254:
8243:
8237:
8236:
8234:
8232:
8227:
8218:
8212:
8211:
8210:
8193:
8187:
8186:
8165:
8159:
8158:
8143:Łukasiewicz, Jan
8139:
8133:
8132:
8128:978-0-41507310-3
8108:
8102:
8101:
8093:
8087:
8086:
8074:
8068:
8067:
8065:
8063:
8049:
8043:
8042:
8022:
8016:
8015:
8005:
7979:
7967:
7961:
7960:
7952:
7942:
7936:
7935:
7925:
7917:
7911:
7909:
7901:
7895:
7894:
7882:
7876:
7875:
7867:
7861:
7860:
7852:
7846:
7845:
7835:
7826:
7825:
7815:
7809:
7808:
7806:
7805:
7786:
7773:
7772:
7750:
7744:
7743:
7723:
7713:
7711:
7700:
7691:
7690:
7688:
7686:
7680:Wolfram Research
7665:
7559:
7555:
7552:
7549:
7547:
7542:
7538:
7535:
7532:
7530:
7525:
7523:
7522:
7517:
7501:
7500:\nleftrightarrow
7465:XOR linked lists
7449:
7448:
7442:
7441:
7435:
7434:
7428:
7427:
7421:
7420:
7414:
7413:
7335:
7333:
7332:
7327:
7274:
7272:
7271:
7266:
7264:
7263:
7254:
7246:
7241:
7211:
7210:
7204:
7203:
7197:
7196:
7171:
7169:
7168:
7163:
7161:
7160:
7151:
7143:
7138:
7112:addition is the
7079:Computer science
7052:
7044:
7042:
7041:
7036:
7022:
7015:
7013:
7012:
7007:
7003:
6997:
6988:
6979:
6977:
6976:
6971:
6967:
6961:
6951:
6949:
6948:
6943:
6929:
6927:
6926:
6921:
6917:
6911:
6902:
6900:
6899:
6894:
6890:
6884:
6875:
6873:
6872:
6867:
6863:
6851:
6842:
6841:
6804:
6796:
6794:
6793:
6788:
6774:
6765:
6763:
6762:
6757:
6737:
6735:
6734:
6729:
6715:
6713:
6712:
6707:
6688:
6687:
6674:
6666:
6664:
6663:
6658:
6644:
6635:
6633:
6632:
6627:
6607:
6605:
6604:
6599:
6585:
6583:
6582:
6577:
6558:
6557:
6544:
6537:
6535:
6534:
6529:
6516:
6508:
6506:
6505:
6500:
6486:
6478:
6476:
6475:
6470:
6456:
6447:
6445:
6444:
6439:
6414:
6412:
6411:
6406:
6393:
6391:
6390:
6385:
6355:
6353:
6352:
6347:
6333:
6331:
6330:
6325:
6306:
6305:
6290:
6282:
6280:
6279:
6274:
6260:
6252:
6250:
6249:
6244:
6230:
6223:
6221:
6220:
6215:
6211:
6205:
6196:
6187:
6185:
6184:
6179:
6175:
6169:
6159:
6157:
6156:
6151:
6137:
6135:
6134:
6129:
6125:
6119:
6109:
6107:
6106:
6101:
6087:
6085:
6084:
6079:
6075:
6069:
6060:
6058:
6057:
6052:
6048:
6042:
6033:
6031:
6030:
6025:
6021:
6015:
6006:
6005:
5986:
5984:
5983:
5978:
5894:
5887:
5885:
5884:
5879:
5875:
5872:
5869:
5863:
5860:
5857:
5848:
5840:
5838:
5837:
5832:
5818:
5810:
5808:
5807:
5802:
5788:
5781:
5779:
5778:
5773:
5769:
5766:
5763:
5757:
5754:
5751:
5742:
5733:
5731:
5730:
5725:
5718:
5709:
5707:
5706:
5701:
5697:
5694:
5691:
5685:
5682:
5679:
5670:
5668:
5667:
5662:
5632:
5630:
5629:
5624:
5610:
5608:
5607:
5602:
5577:
5575:
5574:
5569:
5565:
5562:
5559:
5553:
5550:
5547:
5538:
5536:
5535:
5530:
5523:
5514:
5513:
5498:
5490:
5488:
5487:
5482:
5468:
5459:
5457:
5456:
5451:
5431:
5429:
5428:
5423:
5409:
5407:
5406:
5401:
5382:
5381:
5362:
5360:
5359:
5354:
5346:
5331:
5329:
5328:
5323:
5315:
5300:
5298:
5297:
5292:
5274:
5272:
5271:
5266:
5254:
5252:
5251:
5246:
5178:
5171:
5169:
5168:
5163:
5151:
5149:
5148:
5143:
5123:
5121:
5120:
5115:
5095:
5093:
5092:
5087:
5065:
5063:
5062:
5057:
5035:
5033:
5032:
5027:
5013:
5011:
5010:
5005:
4980:
4978:
4977:
4972:
4958:
4956:
4955:
4950:
4933:
4931:
4930:
4925:
4909:
4907:
4906:
4901:
4885:
4883:
4882:
4877:
4865:
4863:
4862:
4857:
4809:
4807:
4806:
4801:
4787:
4785:
4784:
4779:
4751:
4749:
4748:
4743:
4723:
4721:
4720:
4715:
4701:
4699:
4698:
4693:
4691:
4683:
4674:
4672:
4671:
4666:
4664:
4656:
4647:
4645:
4644:
4639:
4627:
4625:
4624:
4619:
4607:
4605:
4604:
4599:
4587:
4585:
4584:
4579:
4567:
4565:
4564:
4559:
4551:
4543:
4527:
4525:
4524:
4519:
4517:
4509:
4498:
4496:
4495:
4490:
4478:
4476:
4475:
4470:
4458:
4456:
4455:
4450:
4432:
4430:
4429:
4424:
4406:
4404:
4403:
4398:
4382:
4380:
4379:
4374:
4277:
4275:
4274:
4269:
4267:
4266:
4261:
4243:Boolean function
4237:
4235:
4234:
4229:
4227:
4223:
4166:
4099:
4097:
4096:
4091:
4089:
4088:
4083:
4070:
4068:
4067:
4062:
4060:
4059:
4054:
4041:
4039:
4038:
4033:
4021:
4019:
4018:
4013:
3998:
3996:
3995:
3990:
3965:
3963:
3962:
3957:
3955:
3954:
3949:
3934:
3932:
3931:
3926:
3902:
3900:
3899:
3894:
3882:
3880:
3879:
3874:
3855:
3853:
3852:
3847:
3796:
3794:
3793:
3788:
3752:
3750:
3749:
3744:
3701:
3699:
3698:
3693:
3677:
3675:
3674:
3669:
3647:
3645:
3644:
3639:
3578:De Morgan's laws
3567:
3565:
3564:
3559:
3557:
3550:
3545:
3537:
3464:
3457:
3449:
3444:
3436:
3360:
3353:
3345:
3340:
3332:
3224:De Morgan's laws
3218:
3216:
3215:
3210:
3208:
3132:
3130:
3129:
3124:
3122:
3040:
3038:
3037:
3032:
3011:
3009:
3008:
3003:
3001:
2951:
2898:
2791:
2711:
2634:
2632:
2631:
2626:
2614:
2612:
2611:
2606:
2594:
2592:
2591:
2586:
2571:
2569:
2568:
2563:
2561:
2488:
2486:
2485:
2480:
2459:
2457:
2456:
2451:
2449:
2379:
2377:
2376:
2371:
2355:
2353:
2352:
2347:
2331:
2329:
2328:
2323:
2312:("logical and",
2307:
2305:
2304:
2299:
2281:
2279:
2278:
2273:
2265:
2250:
2248:
2247:
2242:
2166:
2164:
2163:
2158:
2139:
2137:
2136:
2131:
2118:
2116:
2115:
2110:
2097:
2096:
2093:
2091:
2090:
2085:
2058:
2036:
2034:
2033:
2028:
2016:
2014:
2013:
2008:
1996:
1994:
1993:
1988:
1975:
1970:
1968:
1967:
1962:
1960:
1943:
1941:
1940:
1935:
1933:
1925:
1916:
1914:
1913:
1908:
1906:
1905:
1897:
1879:
1878:
1875:
1874:
1871:
1868:
1865:
1856:
1855:
1852:
1851:
1848:
1845:
1842:
1839:
1830:
1829:
1826:
1825:
1822:
1818:
1817:
1814:
1811:
1808:
1799:
1798:
1795:
1794:
1791:
1788:
1784:
1783:
1780:
1777:
1774:
1758:
1756:
1755:
1750:
1711:logical operator
1684:
1682:
1681:
1676:
1636:
1629:
1622:
1608:
1607:
1552:Boolean function
1518:Related concepts
1509:
1507:
1506:
1501:
1483:
1481:
1480:
1475:
1457:
1455:
1454:
1449:
1423:
1421:
1420:
1415:
1397:
1395:
1394:
1389:
1384:
1356:
1354:
1353:
1348:
1341:
1340:
1337:
1333:
1314:
1312:
1311:
1306:
1288:
1286:
1285:
1280:
1262:
1260:
1259:
1254:
1236:
1234:
1233:
1228:
1202:
1200:
1199:
1194:
1179:
1177:
1176:
1171:
1169:
1161:
1152:
1150:
1149:
1144:
1129:
1127:
1126:
1121:
1098:
1096:
1095:
1090:
1088:
1083:
1072:
1063:
1061:
1060:
1055:
1037:
1035:
1034:
1029:
1024:
1016:
996:
994:
993:
988:
970:
968:
967:
962:
944:
942:
941:
936:
912:
910:
909:
904:
902:
897:
886:
877:
875:
874:
869:
851:
849:
848:
843:
825:
823:
822:
817:
812:
804:
784:
782:
781:
776:
758:
756:
755:
750:
732:
730:
729:
724:
698:
696:
695:
690:
672:
670:
669:
664:
646:
644:
643:
638:
612:
610:
609:
604:
583:
581:
580:
575:
557:
555:
554:
549:
534:
532:
531:
526:
508:
506:
505:
500:
476:
475:
465:
464:
455:
448:
441:
385:
377:
375:
374:
369:
343:
341:
340:
335:
312:
304:
299:
291:
271:
269:
268:
263:
261:
253:
236:
228:
206:
192:
190:
189:
184:
158:
146:
142:
136:
129:
125:
122:
116:
114:
73:
49:
41:
21:
8801:
8800:
8796:
8795:
8794:
8792:
8791:
8790:
8766:
8765:
8764:
8759:
8751:
8749:
8743:
8726:
8723:
8722:
8706:
8689:
8686:
8685:
8660:
8657:
8656:
8635:
8632:
8631:
8606:
8603:
8602:
8586:
8559:
8556:
8555:
8530:
8527:
8526:
8501:
8498:
8497:
8481:
8464:
8461:
8460:
8435:
8432:
8431:
8406:
8403:
8402:
8381:
8378:
8377:
8361:
8352:
8335:
8332:
8331:
8315:
8309:
8267:
8262:
8252:
8250:
8244:
8240:
8230:
8228:
8225:
8219:
8215:
8194:
8190:
8166:
8162:
8140:
8136:
8129:
8121:. p. 496.
8109:
8105:
8094:
8090:
8075:
8071:
8061:
8059:
8050:
8046:
8023:
8019:
7988:(12): 713–723.
7977:
7968:
7964:
7943:
7939:
7918:
7914:
7902:
7898:
7883:
7879:
7868:
7864:
7853:
7849:
7836:
7829:
7818:
7816:
7812:
7803:
7801:
7787:
7776:
7769:
7751:
7747:
7740:
7709:
7701:
7694:
7684:
7682:
7666:
7662:
7658:
7653:
7649:XOR linked list
7570:
7557:
7553:
7550:
7545:
7544:
7540:
7536:
7533:
7528:
7527:
7511:
7508:
7507:
7499:
7496:
7446:
7444:
7439:
7437:
7432:
7430:
7425:
7423:
7418:
7416:
7411:
7409:
7404:XOR is used in
7377:Feistel network
7346:parity function
7297:
7294:
7293:
7259:
7255:
7250:
7242:
7237:
7232:
7229:
7228:
7208:
7206:
7201:
7199:
7194:
7192:
7156:
7152:
7147:
7139:
7134:
7129:
7126:
7125:
7103:
7097:
7081:
7030:
7027:
7026:
6995:
6992:
6991:
6959:
6956:
6955:
6937:
6934:
6933:
6909:
6906:
6905:
6882:
6879:
6878:
6849:
6846:
6845:
6833:
6825:
6816:
6782:
6779:
6778:
6745:
6742:
6741:
6723:
6720:
6719:
6695:
6692:
6691:
6683:
6652:
6649:
6648:
6615:
6612:
6611:
6593:
6590:
6589:
6565:
6562:
6561:
6553:
6523:
6520:
6519:
6494:
6491:
6490:
6464:
6461:
6460:
6421:
6418:
6417:
6400:
6397:
6396:
6367:
6364:
6363:
6341:
6338:
6337:
6313:
6310:
6309:
6302:
6268:
6265:
6264:
6238:
6235:
6234:
6203:
6200:
6199:
6167:
6164:
6163:
6145:
6142:
6141:
6117:
6114:
6113:
6095:
6092:
6091:
6067:
6064:
6063:
6040:
6037:
6036:
6013:
6010:
6009:
6002:
5918:
5915:
5914:
5906:
5855:
5852:
5851:
5826:
5823:
5822:
5796:
5793:
5792:
5749:
5746:
5745:
5716:
5713:
5712:
5677:
5674:
5673:
5644:
5641:
5640:
5618:
5615:
5614:
5584:
5581:
5580:
5545:
5542:
5541:
5521:
5518:
5517:
5510:
5476:
5473:
5472:
5439:
5436:
5435:
5417:
5414:
5413:
5389:
5386:
5385:
5378:
5370:
5342:
5337:
5334:
5333:
5311:
5306:
5303:
5302:
5280:
5277:
5276:
5260:
5257:
5256:
5240:
5237:
5236:
5176:
5157:
5154:
5153:
5137:
5134:
5133:
5126:Polish notation
5109:
5106:
5105:
5102:Jan Łukasiewicz
5075:
5072:
5071:
5068:prefix operator
5051:
5048:
5047:
5021:
5018:
5017:
4999:
4996:
4995:
4966:
4963:
4962:
4941:
4938:
4937:
4919:
4916:
4915:
4895:
4892:
4891:
4871:
4868:
4867:
4819:
4816:
4815:
4795:
4792:
4791:
4773:
4770:
4769:
4737:
4734:
4733:
4709:
4706:
4705:
4682:
4680:
4677:
4676:
4655:
4653:
4650:
4649:
4633:
4630:
4629:
4613:
4610:
4609:
4593:
4590:
4589:
4573:
4570:
4569:
4542:
4537:
4534:
4533:
4508:
4506:
4503:
4502:
4484:
4481:
4480:
4464:
4461:
4460:
4438:
4435:
4434:
4412:
4409:
4408:
4392:
4389:
4388:
4368:
4365:
4364:
4357:
4288:
4262:
4257:
4256:
4254:
4251:
4250:
4225:
4224:
4208:
4191:
4186:
4168:
4167:
4151:
4134:
4129:
4110:
4108:
4105:
4104:
4084:
4079:
4078:
4076:
4073:
4072:
4055:
4050:
4049:
4047:
4044:
4043:
4027:
4024:
4023:
4007:
4004:
4003:
3972:
3969:
3968:
3950:
3945:
3944:
3942:
3939:
3938:
3908:
3905:
3904:
3888:
3885:
3884:
3868:
3865:
3864:
3817:
3814:
3813:
3758:
3755:
3754:
3714:
3711:
3710:
3687:
3684:
3683:
3663:
3660:
3659:
3653:
3585:
3582:
3581:
3574:
3555:
3554:
3538:
3536:
3516:
3511:
3491:
3486:
3469:
3462:
3461:
3448:
3435:
3415:
3410:
3387:
3382:
3365:
3358:
3357:
3344:
3331:
3326:
3321:
3301:
3296:
3276:
3271:
3258:
3256:
3253:
3252:
3206:
3205:
3164:
3159:
3146:
3144:
3141:
3140:
3120:
3119:
3069:
3064:
3051:
3049:
3046:
3045:
3020:
3017:
3016:
2999:
2998:
2981:
2976:
2956:
2949:
2948:
2931:
2926:
2903:
2896:
2895:
2851:
2846:
2796:
2789:
2788:
2756:
2751:
2716:
2709:
2708:
2688:
2683:
2663:
2658:
2645:
2643:
2640:
2639:
2620:
2617:
2616:
2600:
2597:
2596:
2580:
2577:
2576:
2559:
2558:
2517:
2512:
2499:
2497:
2494:
2493:
2468:
2465:
2464:
2447:
2446:
2408:
2403:
2390:
2388:
2385:
2384:
2365:
2362:
2361:
2341:
2338:
2337:
2336:("logical or",
2317:
2314:
2313:
2287:
2284:
2283:
2261:
2256:
2253:
2252:
2230:
2227:
2226:
2219:
2146:
2143:
2142:
2125:
2122:
2121:
2104:
2101:
2100:
2073:
2070:
2069:
2043:
2022:
2019:
2018:
2002:
1999:
1998:
1982:
1979:
1978:
1973:
1951:
1949:
1946:
1945:
1924:
1922:
1919:
1918:
1896:
1895:
1893:
1890:
1889:
1862:
1858:
1836:
1832:
1819:
1805:
1801:
1785:
1771:
1767:
1761:infix operators
1744:
1741:
1740:
1658:
1655:
1654:
1640:
1599:
1566:
1537:Boolean algebra
1532:Predicate logic
1489:
1486:
1485:
1463:
1460:
1459:
1437:
1434:
1433:
1403:
1400:
1399:
1375:
1370:
1367:
1366:
1336:
1328:
1325:
1324:
1294:
1291:
1290:
1268:
1265:
1264:
1242:
1239:
1238:
1216:
1213:
1212:
1185:
1182:
1181:
1160:
1158:
1155:
1154:
1135:
1132:
1131:
1112:
1109:
1108:
1073:
1071:
1069:
1066:
1065:
1043:
1040:
1039:
1015:
1010:
1007:
1006:
976:
973:
972:
950:
947:
946:
924:
921:
920:
887:
885:
883:
880:
879:
857:
854:
853:
831:
828:
827:
803:
798:
795:
794:
764:
761:
760:
738:
735:
734:
712:
709:
708:
678:
675:
674:
652:
649:
648:
626:
623:
622:
589:
586:
585:
563:
560:
559:
540:
537:
536:
514:
511:
510:
488:
485:
484:
459:
384:Post's lattices
383:
357:
354:
353:
303:
290:
285:
282:
281:
252:
227:
225:
222:
221:
172:
169:
168:
137:
126:
120:
117:
74:
72:
62:
50:
39:
28:
23:
22:
15:
12:
11:
5:
8799:
8789:
8788:
8783:
8778:
8761:
8760:
8748:
8745:
8744:
8742:
8741:
8730:
8711:
8708:
8707:
8705:
8704:
8693:
8675:
8664:
8650:
8639:
8624:Nonimplication
8621:
8610:
8591:
8588:
8587:
8585:
8584:
8581:Digital buffer
8574:
8563:
8545:
8534:
8516:
8505:
8486:
8483:
8482:
8480:
8479:
8468:
8450:
8439:
8421:
8410:
8396:
8385:
8366:
8363:
8362:
8355:
8353:
8351:
8350:
8339:
8320:
8317:
8316:
8308:
8307:
8300:
8293:
8285:
8279:
8278:
8273:
8266:
8265:External links
8263:
8261:
8260:
8238:
8213:
8188:
8160:
8134:
8127:
8103:
8088:
8069:
8044:
8033:(3): 288–309.
8017:
7971:Shannon, C. E.
7962:
7953:Translated as
7937:
7912:
7896:
7877:
7862:
7847:
7827:
7810:
7774:
7767:
7745:
7738:
7714:Translated as
7692:
7659:
7657:
7654:
7652:
7651:
7646:
7641:
7636:
7631:
7626:
7621:
7616:
7611:
7606:
7601:
7596:
7591:
7586:
7581:
7571:
7569:
7566:
7515:
7495:
7492:
7488:alpha channels
7480:bounding boxes
7350:
7349:
7344:returned by a
7338:if and only if
7325:
7322:
7319:
7316:
7313:
7310:
7307:
7304:
7301:
7286:
7283:
7262:
7258:
7253:
7249:
7245:
7240:
7236:
7218:
7217:
7190:
7187:
7184:
7181:
7159:
7155:
7150:
7146:
7142:
7137:
7133:
7099:Main article:
7096:
7093:
7080:
7077:
7058:
7057:
7054:
7053:
7046:
7034:
7023:
7016:
7002:
6989:
6981:
6980:
6966:
6953:
6941:
6930:
6916:
6903:
6889:
6876:
6862:
6859:
6856:
6834:
6831:
6829:
6826:
6819:
6817:
6813:Walsh spectrum
6811:
6809:
6806:
6805:
6798:
6786:
6775:
6767:
6766:
6755:
6752:
6749:
6739:
6727:
6716:
6705:
6702:
6699:
6684:
6681:
6679:
6676:
6675:
6668:
6656:
6645:
6637:
6636:
6625:
6622:
6619:
6609:
6597:
6586:
6575:
6572:
6569:
6554:
6551:
6549:
6546:
6545:
6538:
6527:
6517:
6510:
6498:
6487:
6480:
6468:
6457:
6449:
6448:
6437:
6434:
6431:
6428:
6425:
6415:
6404:
6394:
6383:
6380:
6377:
6374:
6371:
6361:
6359:
6357:
6345:
6334:
6323:
6320:
6317:
6303:
6297:
6295:
6292:
6291:
6284:
6272:
6261:
6254:
6242:
6231:
6224:
6210:
6197:
6189:
6188:
6174:
6161:
6149:
6138:
6124:
6111:
6099:
6088:
6074:
6061:
6047:
6034:
6020:
6003:
5997:
5995:
5976:
5973:
5970:
5967:
5964:
5961:
5958:
5955:
5952:
5949:
5946:
5943:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5907:
5903:Distributivity
5901:
5899:
5896:
5895:
5888:
5868:
5849:
5842:
5830:
5819:
5812:
5800:
5789:
5782:
5762:
5743:
5735:
5734:
5723:
5710:
5690:
5671:
5660:
5657:
5654:
5651:
5648:
5638:
5636:
5634:
5622:
5611:
5600:
5597:
5594:
5591:
5588:
5578:
5558:
5539:
5528:
5511:
5505:
5503:
5500:
5499:
5492:
5480:
5469:
5461:
5460:
5449:
5446:
5443:
5433:
5421:
5410:
5399:
5396:
5393:
5379:
5373:
5369:
5366:
5365:
5364:
5352:
5349:
5345:
5341:
5321:
5318:
5314:
5310:
5290:
5287:
5284:
5264:
5244:
5229:
5188:to denote the
5173:
5161:
5141:
5113:
5096:) was used by
5085:
5082:
5079:
5055:
5045:
5040:, was used by
5025:
5015:
5003:
4983:Claude Shannon
4970:
4960:
4948:
4945:
4935:
4923:
4899:
4875:
4855:
4852:
4849:
4845:
4841:
4838:
4835:
4832:
4829:
4826:
4823:
4812:Giuseppe Peano
4799:
4789:
4777:
4766:Giuseppe Peano
4741:
4730:Ernst Schröder
4728:, was used by
4713:
4703:
4689:
4686:
4662:
4659:
4648:", i.e., used
4637:
4617:
4597:
4577:
4557:
4554:
4549:
4546:
4541:
4515:
4512:
4500:
4488:
4468:
4448:
4445:
4442:
4422:
4419:
4416:
4396:
4372:
4356:
4353:
4319:
4318:
4315:
4300:
4299:
4287:
4284:
4265:
4260:
4239:
4238:
4222:
4219:
4215:
4212:
4207:
4204:
4201:
4198:
4195:
4192:
4190:
4187:
4185:
4182:
4179:
4176:
4173:
4170:
4169:
4165:
4162:
4158:
4155:
4150:
4147:
4144:
4141:
4138:
4135:
4133:
4130:
4128:
4125:
4122:
4119:
4116:
4113:
4112:
4087:
4082:
4058:
4053:
4031:
4011:
3988:
3985:
3982:
3979:
3976:
3953:
3948:
3924:
3921:
3918:
3915:
3912:
3903:over elements
3892:
3872:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3786:
3783:
3780:
3777:
3774:
3771:
3768:
3765:
3762:
3742:
3739:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3691:
3667:
3652:
3649:
3637:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3573:
3570:
3569:
3568:
3553:
3548:
3544:
3541:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3515:
3512:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3490:
3487:
3485:
3482:
3479:
3476:
3473:
3470:
3468:
3465:
3463:
3460:
3455:
3452:
3447:
3442:
3439:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3414:
3411:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3386:
3383:
3381:
3378:
3375:
3372:
3369:
3366:
3364:
3361:
3359:
3356:
3351:
3348:
3343:
3338:
3335:
3330:
3327:
3325:
3322:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3300:
3297:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3275:
3272:
3270:
3267:
3264:
3261:
3260:
3220:
3219:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3171:
3168:
3165:
3163:
3160:
3158:
3155:
3152:
3149:
3148:
3134:
3133:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3068:
3065:
3063:
3060:
3057:
3054:
3053:
3030:
3027:
3024:
3013:
3012:
2997:
2994:
2991:
2988:
2985:
2982:
2980:
2977:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2955:
2952:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2930:
2927:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2902:
2899:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2850:
2847:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2795:
2792:
2790:
2787:
2784:
2781:
2778:
2775:
2772:
2769:
2766:
2763:
2760:
2757:
2755:
2752:
2750:
2747:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2715:
2712:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2687:
2684:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2662:
2659:
2657:
2654:
2651:
2648:
2647:
2624:
2604:
2584:
2573:
2572:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2516:
2513:
2511:
2508:
2505:
2502:
2501:
2478:
2475:
2472:
2461:
2460:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2407:
2404:
2402:
2399:
2396:
2393:
2392:
2380:) as follows:
2369:
2345:
2321:
2297:
2294:
2291:
2271:
2268:
2264:
2260:
2240:
2237:
2234:
2223:if and only if
2218:
2215:
2212:
2211:
2208:
2205:
2201:
2200:
2197:
2194:
2190:
2189:
2186:
2183:
2179:
2178:
2175:
2172:
2168:
2167:
2156:
2153:
2150:
2140:
2129:
2119:
2108:
2083:
2080:
2077:
2042:
2039:
2026:
2006:
1986:
1958:
1955:
1931:
1928:
1903:
1900:
1748:
1728:are true. XOR
1674:
1671:
1668:
1665:
1662:
1642:
1641:
1639:
1638:
1631:
1624:
1616:
1613:
1612:
1601:
1600:
1598:
1597:
1592:
1587:
1582:
1576:
1573:
1572:
1568:
1567:
1565:
1564:
1559:
1554:
1549:
1547:Truth function
1544:
1539:
1534:
1529:
1523:
1520:
1519:
1515:
1514:
1511:
1510:
1499:
1496:
1493:
1473:
1470:
1467:
1447:
1444:
1441:
1431:
1425:
1424:
1413:
1410:
1407:
1387:
1382:
1379:
1374:
1364:
1358:
1357:
1346:
1332:
1322:
1316:
1315:
1304:
1301:
1298:
1278:
1275:
1272:
1252:
1249:
1246:
1226:
1223:
1220:
1210:
1204:
1203:
1192:
1189:
1167:
1164:
1142:
1139:
1119:
1116:
1106:
1100:
1099:
1086:
1082:
1079:
1076:
1053:
1050:
1047:
1027:
1022:
1019:
1014:
1004:
998:
997:
986:
983:
980:
960:
957:
954:
934:
931:
928:
918:
914:
913:
900:
896:
893:
890:
867:
864:
861:
841:
838:
835:
815:
810:
807:
802:
792:
786:
785:
774:
771:
768:
748:
745:
742:
722:
719:
716:
706:
700:
699:
688:
685:
682:
662:
659:
656:
636:
633:
630:
620:
614:
613:
602:
599:
596:
593:
573:
570:
567:
547:
544:
524:
521:
518:
498:
495:
492:
482:
472:
471:
461:
460:
458:
457:
450:
443:
435:
432:
431:
428:
424:
423:
420:
414:
413:
410:
404:
403:
400:
396:
395:
392:
388:
387:
379:
378:
367:
364:
361:
351:
345:
344:
333:
330:
327:
324:
321:
318:
315:
310:
307:
302:
297:
294:
289:
279:
273:
272:
259:
256:
251:
248:
245:
242:
239:
234:
231:
219:
213:
212:
208:
207:
200:
194:
193:
182:
179:
176:
166:
160:
159:
151:
150:
139:
138:
80:"Exclusive or"
53:
51:
44:
26:
9:
6:
4:
3:
2:
8798:
8787:
8784:
8782:
8779:
8777:
8774:
8773:
8771:
8758:
8746:
8720:
8716:
8715:Contradiction
8713:
8712:
8709:
8691:
8683:
8679:
8676:
8662:
8654:
8651:
8637:
8629:
8625:
8622:
8600:
8596:
8593:
8592:
8589:
8582:
8578:
8575:
8553:
8549:
8548:Biconditional
8546:
8532:
8524:
8520:
8517:
8495:
8491:
8488:
8487:
8484:
8466:
8458:
8454:
8451:
8429:
8425:
8422:
8400:
8397:
8375:
8371:
8368:
8367:
8364:
8359:
8329:
8325:
8322:
8321:
8318:
8314:
8306:
8301:
8299:
8294:
8292:
8287:
8286:
8283:
8277:
8274:
8272:
8271:All About XOR
8269:
8268:
8249:
8242:
8224:
8217:
8208:
8207:
8202:
8199:
8192:
8184:
8183:
8178:
8174:
8170:
8164:
8156:
8152:
8148:
8144:
8138:
8130:
8124:
8120:
8116:
8115:
8107:
8099:
8092:
8085:(1): 274–304.
8084:
8080:
8073:
8057:
8056:
8048:
8040:
8036:
8032:
8028:
8021:
8013:
8009:
8004:
7999:
7995:
7991:
7987:
7983:
7976:
7972:
7966:
7958:
7950:
7949:
7941:
7933:
7932:
7926:Reprinted in
7923:
7916:
7907:
7900:
7892:
7888:
7881:
7873:
7866:
7858:
7851:
7843:
7842:
7834:
7832:
7823:
7814:
7799:
7795:
7794:"Disjunction"
7791:
7785:
7783:
7781:
7779:
7770:
7768:9781420070033
7764:
7760:
7756:
7749:
7741:
7735:
7731:
7727:
7722:
7721:
7708:
7707:
7699:
7697:
7681:
7677:
7676:
7671:
7664:
7660:
7650:
7647:
7645:
7642:
7640:
7637:
7635:
7632:
7630:
7627:
7625:
7624:Logical value
7622:
7620:
7619:Logical graph
7617:
7615:
7612:
7610:
7607:
7605:
7602:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7582:
7580:
7576:
7573:
7572:
7565:
7563:
7513:
7505:
7491:
7489:
7485:
7481:
7477:
7472:
7470:
7466:
7462:
7460:
7455:
7451:
7407:
7402:
7400:
7396:
7395:entropy pools
7391:
7389:
7385:
7380:
7378:
7374:
7370:
7365:
7361:
7359:
7355:
7347:
7343:
7339:
7323:
7320:
7317:
7314:
7311:
7308:
7305:
7302:
7299:
7291:
7287:
7284:
7281:
7280:
7279:
7276:
7260:
7247:
7243:
7227:
7223:
7215:
7191:
7188:
7185:
7182:
7179:
7178:
7177:
7157:
7144:
7140:
7123:
7119:
7115:
7111:
7107:
7102:
7090:
7085:
7076:
7074:
7071:
7067:
7063:
7051:
7047:
7024:
7021:
7017:
7000:
6990:
6987:
6983:
6982:
6964:
6954:
6931:
6914:
6904:
6887:
6877:
6860:
6857:
6854:
6844:
6843:
6839:
6835:
6827:
6823:
6814:
6803:
6799:
6776:
6773:
6769:
6768:
6753:
6750:
6747:
6740:
6717:
6703:
6700:
6697:
6690:
6689:
6685:
6673:
6669:
6654:
6646:
6643:
6639:
6638:
6623:
6620:
6617:
6610:
6595:
6587:
6573:
6570:
6567:
6560:
6559:
6555:
6543:
6539:
6518:
6515:
6511:
6488:
6485:
6481:
6466:
6458:
6455:
6451:
6450:
6432:
6429:
6426:
6416:
6395:
6378:
6375:
6372:
6362:
6360:
6358:
6343:
6335:
6321:
6315:
6308:
6307:
6304:
6300:
6289:
6285:
6270:
6262:
6259:
6255:
6232:
6229:
6225:
6208:
6198:
6195:
6191:
6190:
6172:
6162:
6147:
6139:
6122:
6112:
6089:
6072:
6062:
6045:
6035:
6018:
6008:
6007:
6004:
6000:
5993:
5990:
5971:
5968:
5965:
5959:
5953:
5950:
5947:
5941:
5935:
5932:
5929:
5923:
5920:
5912:
5908:
5904:
5893:
5889:
5866:
5850:
5847:
5843:
5820:
5817:
5813:
5790:
5787:
5783:
5760:
5744:
5741:
5737:
5736:
5721:
5711:
5688:
5672:
5655:
5652:
5649:
5639:
5637:
5635:
5612:
5595:
5592:
5589:
5579:
5556:
5540:
5526:
5516:
5515:
5512:
5508:
5507:Associativity
5497:
5493:
5470:
5467:
5463:
5462:
5447:
5444:
5441:
5434:
5411:
5397:
5394:
5391:
5384:
5383:
5380:
5376:
5375:Commutativity
5371:
5350:
5347:
5343:
5339:
5319:
5316:
5312:
5308:
5288:
5285:
5282:
5262:
5242:
5234:
5230:
5227:
5223:
5219:
5215:
5211:
5207:
5203:
5199:
5195:
5191:
5187:
5183:
5179:
5174:
5159:
5139:
5131:
5127:
5111:
5103:
5099:
5083:
5080:
5077:
5069:
5053:
5046:
5043:
5042:Alonzo Church
5039:
5023:
5016:
5001:
4992:
4988:
4984:
4968:
4961:
4946:
4943:
4936:
4921:
4913:
4897:
4889:
4873:
4853:
4850:
4847:
4843:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4813:
4797:
4790:
4775:
4767:
4763:
4759:
4755:
4739:
4731:
4727:
4711:
4704:
4684:
4657:
4635:
4615:
4595:
4575:
4555:
4552:
4544:
4539:
4531:
4510:
4501:
4486:
4466:
4446:
4443:
4440:
4420:
4417:
4414:
4394:
4386:
4370:
4363:
4362:
4361:
4352:
4350:
4345:
4343:
4339:
4335:
4331:
4327:
4324:
4316:
4313:
4312:
4311:
4309:
4305:
4297:
4296:
4295:
4293:
4283:
4281:
4263:
4248:
4244:
4217:
4213:
4205:
4202:
4199:
4196:
4193:
4183:
4180:
4177:
4174:
4171:
4160:
4156:
4148:
4145:
4142:
4139:
4136:
4126:
4123:
4120:
4117:
4114:
4103:
4102:
4101:
4085:
4056:
4029:
4009:
4000:
3983:
3980:
3977:
3966:
3951:
3919:
3916:
3913:
3890:
3870:
3862:
3861:abelian group
3858:
3840:
3837:
3831:
3828:
3825:
3810:
3808:
3804:
3800:
3781:
3778:
3772:
3769:
3766:
3737:
3734:
3728:
3725:
3722:
3707:
3705:
3689:
3681:
3665:
3658:
3655:Although the
3648:
3635:
3632:
3626:
3623:
3617:
3614:
3611:
3599:
3596:
3593:
3579:
3542:
3539:
3527:
3524:
3521:
3513:
3505:
3502:
3499:
3488:
3480:
3477:
3474:
3466:
3450:
3445:
3437:
3426:
3423:
3420:
3412:
3404:
3398:
3395:
3384:
3376:
3373:
3370:
3362:
3354:
3346:
3341:
3333:
3328:
3323:
3315:
3312:
3309:
3298:
3290:
3284:
3281:
3273:
3268:
3265:
3262:
3251:
3250:
3249:
3246:
3244:
3240:
3236:
3232:
3227:
3225:
3199:
3193:
3190:
3181:
3175:
3172:
3169:
3161:
3156:
3153:
3150:
3139:
3138:
3137:
3110:
3104:
3101:
3092:
3086:
3083:
3080:
3066:
3061:
3058:
3055:
3044:
3043:
3042:
3028:
3025:
3022:
2992:
2989:
2986:
2978:
2970:
2967:
2964:
2953:
2942:
2939:
2936:
2928:
2920:
2914:
2911:
2900:
2886:
2883:
2880:
2871:
2865:
2862:
2859:
2848:
2837:
2831:
2828:
2819:
2813:
2807:
2804:
2793:
2782:
2779:
2773:
2767:
2764:
2753:
2745:
2739:
2733:
2727:
2724:
2713:
2702:
2699:
2696:
2685:
2677:
2671:
2668:
2660:
2655:
2652:
2649:
2638:
2637:
2636:
2622:
2602:
2552:
2549:
2546:
2537:
2531:
2525:
2522:
2514:
2509:
2506:
2503:
2492:
2491:
2490:
2476:
2473:
2470:
2440:
2437:
2434:
2425:
2419:
2416:
2413:
2405:
2400:
2397:
2394:
2383:
2382:
2381:
2359:
2343:
2335:
2319:
2311:
2295:
2292:
2289:
2269:
2266:
2262:
2258:
2238:
2235:
2232:
2224:
2209:
2206:
2203:
2202:
2198:
2195:
2192:
2191:
2187:
2184:
2181:
2180:
2176:
2173:
2170:
2169:
2154:
2151:
2148:
2141:
2127:
2120:
2106:
2099:
2098:
2095:
2081:
2078:
2075:
2067:
2056:
2052:
2047:
2038:
2024:
2004:
1984:
1976:
1956:
1953:
1926:
1901:
1898:
1887:
1883:
1877:
1854:
1828:
1797:
1765:
1762:
1746:
1738:
1733:
1731:
1727:
1722:
1720:
1716:
1712:
1708:
1707:
1702:
1698:
1694:
1690:
1672:
1669:
1666:
1663:
1660:
1652:
1648:
1637:
1632:
1630:
1625:
1623:
1618:
1617:
1615:
1614:
1611:
1603:
1602:
1596:
1593:
1591:
1588:
1586:
1583:
1581:
1580:Digital logic
1578:
1577:
1575:
1574:
1570:
1569:
1563:
1562:Scope (logic)
1560:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1528:
1525:
1524:
1522:
1521:
1517:
1516:
1497:
1491:
1471:
1468:
1465:
1445:
1439:
1432:
1430:
1427:
1426:
1411:
1408:
1405:
1385:
1380:
1377:
1372:
1365:
1363:
1360:
1359:
1344:
1330:
1323:
1321:
1318:
1317:
1302:
1299:
1296:
1276:
1273:
1270:
1250:
1247:
1244:
1224:
1221:
1218:
1211:
1209:
1206:
1205:
1190:
1187:
1162:
1140:
1137:
1117:
1107:
1105:
1102:
1101:
1080:
1077:
1074:
1051:
1045:
1025:
1017:
1012:
1005:
1003:
1000:
999:
984:
981:
978:
958:
955:
952:
932:
929:
926:
919:
917:nonequivalent
916:
915:
894:
891:
888:
865:
862:
859:
839:
833:
813:
805:
800:
793:
791:
788:
787:
772:
766:
746:
743:
740:
720:
714:
707:
705:
702:
701:
686:
680:
660:
654:
634:
631:
628:
621:
619:
616:
615:
600:
591:
571:
565:
545:
542:
522:
519:
516:
496:
493:
490:
483:
481:
478:
477:
474:
473:
470:
467:
466:
456:
451:
449:
444:
442:
437:
436:
433:
429:
425:
421:
419:
415:
411:
409:
405:
401:
397:
393:
389:
386:
380:
365:
362:
359:
352:
350:
346:
328:
325:
322:
316:
305:
300:
292:
280:
278:
274:
254:
249:
246:
243:
240:
237:
229:
220:
218:
214:
209:
205:
201:
199:
195:
177:
167:
165:
161:
157:
152:
147:
135:
132:
124:
113:
110:
106:
103:
99:
96:
92:
89:
85:
82: –
81:
77:
76:Find sources:
70:
66:
60:
59:
54:This article
52:
48:
43:
42:
37:
33:
19:
8595:Joint denial
8519:Exclusive or
8518:
8251:. Retrieved
8241:
8229:. Retrieved
8216:
8204:
8191:
8181:
8163:
8150:
8146:
8137:
8113:
8106:
8097:
8091:
8082:
8078:
8072:
8060:. Retrieved
8054:
8047:
8030:
8026:
8020:
8003:1721.1/11173
7985:
7981:
7965:
7956:
7947:
7940:
7930:
7921:
7915:
7905:
7899:
7890:
7880:
7871:
7865:
7856:
7850:
7840:
7821:
7813:
7802:. Retrieved
7797:
7790:Aloni, Maria
7758:
7748:
7719:
7705:
7683:. Retrieved
7673:
7663:
7604:Inclusive or
7554:CIRCLED PLUS
7541:⊻
7497:
7473:
7463:
7456:
7452:
7403:
7392:
7381:
7373:one-time pad
7369:cryptography
7366:
7362:
7351:
7277:
7226:vector space
7221:
7219:
7175:
7114:exclusive or
7113:
7066:exclusive or
7065:
7059:
6815:: (2,0,0,−2)
6299:Monotonicity
5235:of two sets
5175:
4981:was used by
4911:
4887:
4810:was used by
4762:Hugh MacColl
4754:George Boole
4568:to express "
4528:was used by
4385:George Boole
4383:was used by
4358:
4349:soit... soit
4348:
4346:
4320:
4304:felicitously
4301:
4289:
4240:
4001:
3856:
3811:
3709:The systems
3708:
3654:
3575:
3247:
3228:
3221:
3135:
3014:
2574:
2462:
2220:
2063:
2051:Walsh matrix
1972:
1885:
1881:
1763:
1734:
1729:
1723:
1704:
1700:
1696:
1692:
1689:Exclusive or
1688:
1687:
1651:Venn diagram
1571:Applications
1361:
399:1-preserving
391:0-preserving
211:Normal forms
144:Exclusive or
127:
118:
108:
101:
94:
87:
75:
63:Please help
58:verification
55:
8776:Dichotomies
8678:Conjunction
8628:NIMPLY gate
8453:Disjunction
8424:Implication
7189:0 XOR 0 = 0
7186:0 XOR 1 = 1
7183:1 XOR 0 = 1
7180:1 XOR 1 = 0
6832:Involution:
5999:Idempotency
5130:connectives
5038:equivalence
4890:; the sign
4866:. The sign
4726:equivalence
4334:cancellable
4022:with 0 and
3704:disjunction
3680:conjunction
2356:), and the
2334:disjunction
2066:truth table
1759:and by the
1542:Truth table
277:Conjunctive
217:Disjunctive
164:Truth table
8770:Categories
8428:IMPLY gate
7804:2020-09-03
7639:XOR cipher
7609:Involution
7390:function.
7342:parity bit
7089:logic gate
6838:involution
5368:Properties
4814:in 1894: "
4342:entailment
4247:polynomial
3580:, we get:
3239:antecedent
2041:Definition
1737:symbolized
618:equivalent
198:Logic gate
91:newspapers
8786:Semantics
8729:⊥
8692:∧
8663:↚
8638:↛
8609:↓
8577:Statement
8562:↔
8552:XNOR gate
8504:¬
8467:∨
8438:→
8409:←
8384:↑
8374:NAND gate
8338:⊤
8324:Tautology
8231:28 August
8206:MathWorld
7675:MathWorld
7579:(Paradox)
7514:↮
7494:Encodings
7436:is lost,
7321:⊕
7315:⊕
7309:⊕
7303:⊕
7060:If using
7033:⇔
7001:⊕
6940:⇔
6888:⊕
6858:⊕
6822:linearity
6785:⇒
6751:∨
6726:⇒
6701:⊕
6655:⇏
6621:⊕
6596:⇏
6571:∧
6526:→
6497:⇔
6467:⇏
6430:⊕
6403:→
6376:⊕
6344:⇏
6319:→
6271:⇎
6241:⇔
6209:⊕
6148:⇎
6098:⇔
6046:⊕
5969:∧
5960:⊕
5951:∧
5933:⊕
5924:∧
5867:⊕
5829:⇔
5799:⇔
5761:⊕
5689:⊕
5653:⊕
5621:⇔
5593:⊕
5557:⊕
5479:⇔
5445:⊕
5420:⇔
5395:⊕
5348:
5344:△
5317:
5313:▽
5286:⊖
5084:ψ
5081:ϕ
4969:⊕
4947:∨
4944:∨
4922:∪
4898:∪
4874:∘
4851:−
4844:∪
4837:−
4825:∘
4798:∘
4688:¯
4685:∨
4661:¯
4658:∨
4608:" or "No
4553:
4548:¯
4545:∨
4514:¯
4511:∨
4487:∨
4330:semantics
4323:pragmatic
4189:⇔
4181:⊕
4146:⋅
4132:⇔
4124:∧
3984:∨
3978:∧
3891:⊕
3871:∧
3841:⊕
3782:∨
3738:∧
3690:∨
3666:∧
3657:operators
3630:¬
3627:↮
3621:⇔
3615:↮
3609:¬
3606:⇔
3597:↮
3588:¬
3547:¯
3503:∧
3494:¬
3489:∧
3478:∨
3454:¯
3441:¯
3402:¬
3399:∨
3393:¬
3385:∧
3374:∨
3350:¯
3337:¯
3313:∧
3307:¬
3299:∨
3288:¬
3285:∧
3266:↮
3197:¬
3194:∨
3188:¬
3182:∧
3173:∨
3154:↮
3108:¬
3105:∧
3099:¬
3093:∨
3084:∧
3072:¬
3059:↮
3026:↮
2990:∨
2979:∧
2968:∧
2959:¬
2940:∨
2929:∧
2918:¬
2915:∨
2909:¬
2884:∨
2878:¬
2872:∧
2863:∨
2849:∧
2835:¬
2832:∨
2826:¬
2820:∧
2811:¬
2808:∨
2780:∨
2771:¬
2768:∧
2754:∧
2743:¬
2740:∨
2731:¬
2728:∧
2700:∧
2694:¬
2686:∨
2675:¬
2672:∧
2653:↮
2623:∨
2603:∧
2583:¬
2550:∧
2544:¬
2538:∨
2529:¬
2526:∧
2507:↮
2474:↮
2438:∧
2429:¬
2426:∧
2417:∨
2398:↮
2368:¬
2344:∨
2320:∧
2267:
2236:↮
2152:⊕
2079:⊕
2005:↮
1985:⊕
1957:_
1954:∨
1930:¯
1927:∨
1902:˙
1899:∨
1670:⊕
1664:⊕
1495:←
1469:⊂
1443:⇐
1409:⊕
1381:_
1378:∨
1300:∥
1274:∣
1222:∨
1188:∼
1166:¯
1138:−
1115:¬
1085:¯
1049:↓
1021:¯
1018:∨
982:↮
899:¯
892:⋅
863:∣
837:↑
809:¯
806:∧
770:→
744:⊃
718:⇒
684:⇋
658:⇔
632:≡
598:&
595:&
569:&
520:⋅
494:∧
427:Self-dual
363:⊕
317:⋅
309:¯
296:¯
258:¯
250:⋅
238:⋅
233:¯
8682:AND gate
8599:NOR gate
8533:↮
8523:XOR gate
8494:NOT gate
8490:Negation
8253:23 March
8175:(1978).
8145:(1929).
8012:51638483
7973:(1938).
7792:(2016).
7644:XOR gate
7568:See also
7551:⊕
7534:⊻
7445:11110000
7438:10011100
7431:01101100
7424:11110000
7417:01101100
7410:10011100
7358:XOR gate
7336:is true
7070:addition
5044:in 1944.
5024:≢
4776:≠
4712:≠
2358:negation
2055:variadic
2025:≢
1730:excludes
1726:operands
1610:Category
1429:converse
956:⇎
930:≢
408:Monotone
121:May 2013
32:XOR gate
8684:)
8680: (
8630:)
8626: (
8601:)
8597: (
8579: (
8554:)
8550: (
8525:)
8521: (
8496:)
8492: (
8459:)
8457:OR gate
8455: (
8430:)
8426: (
8376:)
8372: (
8311:Common
7685:17 June
7634:Rule 90
7589:Ampheck
7484:cursors
5190:bitwise
4588:is-not
3799:monoids
2332:), the
704:implies
105:scholar
8721:
8655:
8401:
8330:
8125:
8062:7 July
8010:
7765:
7736:
7548:
7546:U+2295
7543:) and
7531:
7529:U+22BB
7122:binary
7110:Nimber
7073:modulo
7062:binary
7004:
6998:
6968:
6962:
6918:
6912:
6891:
6885:
6864:
6852:
6212:
6206:
6176:
6170:
6126:
6120:
6076:
6070:
6049:
6043:
6022:
6016:
5876:
5873:
5870:
5864:
5861:
5858:
5770:
5767:
5764:
5758:
5755:
5752:
5719:
5698:
5695:
5692:
5686:
5683:
5680:
5566:
5563:
5560:
5554:
5551:
5548:
5524:
5226:Python
5180:, the
5066:(as a
3682:) and
2017:, and
1735:It is
1342:
1334:
418:Affine
107:
100:
93:
86:
78:
8719:False
8226:(PDF)
8149:[
8008:S2CID
7978:(PDF)
7710:(PDF)
7670:"XOR"
7656:Notes
7504:LaTeX
7502:) in
7354:adder
7214:carry
5992:GF(2)
5989:field
5509:: yes
5377:: yes
5332:, or
5182:caret
4245:as a
3803:group
1709:is a
1703:, or
112:JSTOR
98:books
8328:True
8255:2017
8233:2013
8123:ISBN
8064:2023
7763:ISBN
7734:ISBN
7687:2015
7482:and
7443:and
7415:and
7406:RAID
7397:for
7207:0111
7200:1001
7198:XOR
7193:1110
6820:Non-
6301:: no
6001:: no
5255:and
5231:The
5224:and
5218:Ruby
5214:Perl
5210:Java
3883:and
3797:are
3753:and
3136:or:
2615:and
2064:The
1886:EXOR
1338:XNOR
1320:XNOR
790:NAND
178:0110
84:news
8035:doi
7998:hdl
7990:doi
7726:doi
7537:XOR
7474:In
7375:or
7367:In
7290:odd
7120:in
7116:of
7075:2.
6824:: 0
5222:PHP
5198:C++
4912:vel
4910:to
4888:aut
4628:is
4249:in
4214:mod
4157:mod
3859:an
2282:or
2068:of
1882:EOR
1880:),
1857:or
1764:XOR
1719:odd
1653:of
1362:XOR
1104:NOT
1002:NOR
480:AND
422:yes
394:yes
149:XOR
67:by
8772::
8203:.
8179:.
8171:;
8117:.
8083:35
8081:.
8029:.
8006:.
7996:.
7986:57
7984:.
7980:.
7830:^
7777:^
7757:.
7732:.
7695:^
7678:.
7672:.
7577:•
7564:.
7275:.
7205:=
5913:.
5301:,
5220:,
5216:,
5212:,
5208:,
5204:,
5202:C#
5200:,
5070:,
4764:,
4760:,
4351:.
4282:.
4100::
3857:is
3809:.
3245:.
2037:.
1997:,
1977:,
1971:,
1944:,
1917:,
1888:,
1884:,
1873:ɔː
1847:ɔː
1831:,
1824:ɔː
1800:,
1790:ɔː
1721:.
1699:,
1695:,
1691:,
1484:,
1458:,
1398:,
1289:,
1263:,
1237:,
1208:OR
1180:,
1153:,
1130:,
1064:,
1038:,
971:,
945:,
878:,
852:,
826:,
759:,
733:,
673:,
647:,
584:,
558:,
535:,
509:,
430:no
412:no
402:no
8717:/
8583:)
8326:/
8304:e
8297:t
8290:v
8257:.
8235:.
8209:.
8157:.
8131:.
8066:.
8041:.
8037::
8031:5
8014:.
8000::
7992::
7807:.
7771:.
7742:.
7728::
7689:.
7556:(
7539:(
7447:2
7440:2
7433:2
7426:2
7419:2
7412:2
7348:.
7324:E
7318:D
7312:C
7306:B
7300:A
7261:n
7257:)
7252:Z
7248:2
7244:/
7239:Z
7235:(
7222:n
7216:)
7209:2
7202:2
7195:2
7172:.
7158:4
7154:)
7149:Z
7145:2
7141:/
7136:Z
7132:(
6965:A
6915:B
6861:B
6855:A
6754:B
6748:A
6704:B
6698:A
6624:B
6618:A
6574:B
6568:A
6436:)
6433:C
6427:B
6424:(
6382:)
6379:C
6373:A
6370:(
6322:B
6316:A
6173:A
6123:0
6073:A
6019:A
5975:)
5972:B
5966:C
5963:(
5957:)
5954:A
5948:C
5945:(
5942:=
5939:)
5936:B
5930:A
5927:(
5921:C
5905::
5722:C
5659:)
5656:B
5650:A
5647:(
5599:)
5596:C
5590:B
5587:(
5527:A
5448:A
5442:B
5398:B
5392:A
5363:.
5351:T
5340:S
5320:T
5309:S
5289:T
5283:S
5263:T
5243:S
5228:.
5206:D
5194:C
5177:^
5160:J
5140:J
5112:J
5078:J
5054:J
5002:+
4854:a
4848:b
4840:b
4834:a
4831:=
4828:b
4822:a
4740:=
4636:B
4616:A
4596:B
4576:A
4556:B
4540:A
4467:+
4447:y
4444:+
4441:x
4421:y
4418:,
4415:x
4395:+
4371:+
4264:2
4259:F
4221:)
4218:2
4211:(
4206:q
4203:+
4200:p
4197:=
4194:r
4184:q
4178:p
4175:=
4172:r
4164:)
4161:2
4154:(
4149:q
4143:p
4140:=
4137:r
4127:q
4121:p
4118:=
4115:r
4086:2
4081:F
4057:2
4052:F
4030:T
4010:F
3987:)
3981:,
3975:(
3952:2
3947:F
3923:}
3920:F
3917:,
3914:T
3911:{
3844:)
3838:,
3835:}
3832:F
3829:,
3826:T
3823:{
3820:(
3785:)
3779:,
3776:}
3773:F
3770:,
3767:T
3764:{
3761:(
3741:)
3735:,
3732:}
3729:F
3726:,
3723:T
3720:{
3717:(
3702:(
3678:(
3636:.
3633:q
3624:p
3618:q
3612:p
3603:)
3600:q
3594:p
3591:(
3552:)
3543:q
3540:p
3534:(
3531:)
3528:q
3525:+
3522:p
3519:(
3514:=
3509:)
3506:q
3500:p
3497:(
3484:)
3481:q
3475:p
3472:(
3467:=
3459:)
3451:q
3446:+
3438:p
3433:(
3430:)
3427:q
3424:+
3421:p
3418:(
3413:=
3408:)
3405:q
3396:p
3390:(
3380:)
3377:q
3371:p
3368:(
3363:=
3355:q
3347:p
3342:+
3334:q
3329:p
3324:=
3319:)
3316:q
3310:p
3304:(
3294:)
3291:q
3282:p
3279:(
3274:=
3269:q
3263:p
3203:)
3200:q
3191:p
3185:(
3179:)
3176:q
3170:p
3167:(
3162:=
3157:q
3151:p
3117:)
3114:)
3111:q
3102:p
3096:(
3090:)
3087:q
3081:p
3078:(
3075:(
3067:=
3062:q
3056:p
3029:q
3023:p
2996:)
2993:q
2987:p
2984:(
2974:)
2971:q
2965:p
2962:(
2954:=
2946:)
2943:q
2937:p
2934:(
2924:)
2921:q
2912:p
2906:(
2901:=
2893:)
2890:)
2887:q
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2869:)
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2860:p
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2854:(
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2814:p
2805:p
2802:(
2799:(
2794:=
2786:)
2783:q
2777:)
2774:q
2765:p
2762:(
2759:(
2749:)
2746:p
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