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Binary operations are sometimes written using prefix or (more frequently) postfix notation, both of which dispense with parentheses. They are also called, respectively,
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is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.
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of a binary operation expresses the existence of a result for the operation given any pair of operands.
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is a binary operation since the composition of the two functions is again a function on the set
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The first three examples above are commutative and all of the above examples are associative.
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3391: â Mathematical operation that combines three elements to produce another element
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2894:. Powers are usually also written without operator, but with the second argument as
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3476:(2nd ed.). Springer Science & Business Media. Chapter 2. Partial algebras.
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3382: â Construct associated with a mathematical operation in computer programs
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188:. Other examples are readily found in different areas of mathematics, such as
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is a binary operation since the sum of two real numbers is a real number.
181:
120:
3353:. It depends on authors whether it is considered as a binary operation.
1455:
Many binary operations of interest in both algebra and formal logic are
203:
A binary function that involves several sets is sometimes also called a
4161:
4016:
3987:
3793:
1284:{\displaystyle f(h_{1},h_{2})(c)=(h_{1}\circ h_{2})(c)=h_{1}(h_{2}(c))}
139:) to produce another element. More formally, a binary operation is an
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1832:, is a binary operation which is not commutative since, in general,
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562:
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is a binary operation since the product of two such matrices is a
508:, binary operations are required to be defined on all elements of
4894:
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227:
2562:
is any negative integer. For either set, this operation has a
610:
235:
879:
is a binary operation since the sum of two such matrices is a
4438:
3784:
3629:
144:
3370: â Repeated application of an operation to a sequence
3565:
2962:
3534:
Applied
Algebra: Codes, Ciphers and Discrete Algorithms
3384:
Pages displaying short descriptions of redirect targets
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takes a scalar and a vector to produce a vector, and
96:
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36:
3402: â Mathematical operation with only one operand
3376: â Algebraic structure with a binary operation
449:is a partial binary operation, because one can not
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1870:. It is also not associative, since, in general,
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376:{\displaystyle \,f\colon S\times S\rightarrow S.}
172:are the same set. Examples include the familiar
5344:
2859:rather than by functional notation of the form
1147:{\displaystyle f\colon S\times S\rightarrow S}
561:Typical examples of binary operations are the
3600:
549:, the term binary operation is used for any
3527:
3431:
2465:{\displaystyle f(2,3^{2})=f(2,9)=2^{9}=512}
2223:{\displaystyle f(f(a,b),c)\neq f(a,f(b,c))}
222:Binary operations are the keystone of most
3792:
3607:
3593:
3461:
3179:
2750:Binary operations are often written using
2383:{\displaystyle f(2^{3},2)=f(8,2)=8^{2}=64}
556:
3536:, Upper Saddle River, NJ: Prentice-Hall,
3505:(2nd ed.), Reading: Addison-Wesley,
2502:
2480:
2034:
1767:
928:
789:
705:
633:
348:
3500:
3443:
3363:Category:Properties of binary operations
262:More precisely, a binary operation on a
25:
16:Mathematical operation with two operands
3554:(2nd ed.), Boston: Allyn and Bacon
1658:{\displaystyle f(f(a,b),c)=f(a,f(b,c))}
1317:, the composition of the two functions
219:takes two vectors to produce a scalar.
5345:
3614:
3549:
3419:
2963:Binary operations as ternary relations
1109:{\displaystyle h\colon C\rightarrow C}
50:is a rule for combining the arguments
3588:
3566:
3552:The Theory of Groups: An Introduction
2144:), and is also not associative since
3518:
3455:
3395:Truth table § Binary operations
1925:{\displaystyle a-(b-c)\neq (a-b)-c}
480:is undefined for every real number
13:
3503:A First Course in Abstract Algebra
2735:{\displaystyle \uparrow \uparrow }
14:
5364:
3559:
3109:{\displaystyle S\times S\times S}
938:{\displaystyle M(2,\mathbb {R} )}
799:{\displaystyle M(2,\mathbb {R} )}
5326:
2133:{\displaystyle a^{b}\neq b^{a}}
621:on a single set. For instance,
3449:
3437:
3425:
3413:
3071:
3068:
3056:
3038:
3031:, that is, the set of triples
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2419:
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2318:
2217:
2214:
2202:
2190:
2181:
2172:
2160:
2154:
2074:
2062:
2026:On the set of natural numbers
1995:
1983:
1957:
1945:
1913:
1901:
1895:
1883:
1807:
1795:
1652:
1649:
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1164:
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741:
729:
697:On the set of natural numbers
669:
657:
473:{\displaystyle {\frac {a}{0}}}
364:
257:
1:
5287:History of mathematical logic
3494:
1508:{\displaystyle f(a,b)=f(b,a)}
542:to allow partial operations.
445:. For instance, division of
5212:Primitive recursive function
3077:{\displaystyle (a,b,f(a,b))}
2685:{\displaystyle f(1,b)\neq b}
2647:in the set, which is not an
2509:{\displaystyle \mathbb {Z} }
2487:{\displaystyle \mathbb {N} }
2100:, is not commutative since,
2093:{\displaystyle f(a,b)=a^{b}}
2041:{\displaystyle \mathbb {N} }
1774:{\displaystyle \mathbb {R} }
1084:be the set of all functions
971:matrices with real entries,
832:matrices with real entries,
712:{\displaystyle \mathbb {N} }
640:{\displaystyle \mathbb {R} }
131:is a rule for combining two
7:
3519:Hall, Marshall Jr. (1959),
3356:
2745:
2651:(two sided identity) since
1863:{\displaystyle a-b\neq b-a}
1759:On the set of real numbers
625:On the set of real numbers
10:
5369:
4276:SchröderâBernstein theorem
4003:Monadic predicate calculus
3662:Foundations of mathematics
3550:Rotman, Joseph J. (1973),
3501:Fraleigh, John B. (1976),
3468:George A. GrÀtzer (2008).
2016:{\displaystyle (1-2)-3=-4}
1825:{\displaystyle f(a,b)=a-b}
872:{\displaystyle f(A,B)=A+B}
759:{\displaystyle f(a,b)=a+b}
687:{\displaystyle f(a,b)=a+b}
18:
5322:
5309:Philosophy of mathematics
5258:Automated theorem proving
5240:
5135:
4967:
4860:
4712:
4429:
4405:
4383:Von NeumannâBernaysâGödel
4328:
4222:
4126:
4024:
4015:
3942:
3877:
3783:
3705:
3622:
3368:Iterated binary operation
3266:{\displaystyle S\times S}
1969:{\displaystyle 1-(2-3)=2}
1034:{\displaystyle 2\times 2}
1008:{\displaystyle f(A,B)=AB}
964:{\displaystyle 2\times 2}
898:{\displaystyle 2\times 2}
825:{\displaystyle 2\times 2}
545:Sometimes, especially in
527:{\displaystyle S\times S}
311:{\displaystyle S\times S}
3434:, pg. 176, Definition 67
3406:
2825:{\displaystyle a\cdot b}
2620:{\displaystyle f(a,1)=a}
619:composition of functions
443:partial binary operation
109:{\displaystyle x\circ y}
19:Not to be confused with
4959:Self-verifying theories
4780:Tarski's axiomatization
3731:Tarski's undefinability
3726:incompleteness theorems
3432:Hardy & Walker 2002
3333:is a vector space over
3180:Other binary operations
2952:{\displaystyle ab\ast }
2932:reverse Polish notation
2923:{\displaystyle \ast ab}
2773:{\displaystyle a\ast b}
2494:to the set of integers
2472:. By changing the set
2048:, the binary operation
602:{\displaystyle \times }
557:Properties and examples
289:of the elements of the
5333:Mathematics portal
4944:Proof of impossibility
4592:propositional variable
3902:Propositional calculus
3380:Operator (programming)
3347:
3327:
3307:
3287:
3267:
3230:
3206:
3170:
3150:
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3001:
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2924:
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2887:{\displaystyle f(a,b)}
2853:
2826:
2800:
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2736:
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2641:
2621:
2580:
2556:
2536:
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2466:
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2276:
2250:
2230:. For instance, with
2224:
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2017:
1970:
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1775:
1739:
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1431:(that is, a member of
1425:
1405:
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1338:
1311:
1310:{\displaystyle c\in C}
1285:
1148:
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116:
110:
84:
64:
44:
43:{\displaystyle \circ }
5202:Kolmogorov complexity
5155:Computably enumerable
5055:Model complete theory
4847:Principia Mathematica
3907:Propositional formula
3736:BanachâTarski paradox
3523:, New York: Macmillan
3348:
3328:
3308:
3288:
3268:
3231:
3207:
3186:scalar multiplication
3171:
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3079:
3026:
3002:
2982:
2954:
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2854:
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2713:
2711:{\displaystyle \div }
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1366:
1364:{\displaystyle h_{2}}
1339:
1337:{\displaystyle h_{1}}
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495:
475:
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378:
333:
313:
280:
209:scalar multiplication
198:conjugation in groups
194:matrix multiplication
174:arithmetic operations
150:More specifically, a
111:
85:
65:
45:
29:
5150:ChurchâTuring thesis
5137:Computability theory
4346:continuum hypothesis
3864:Square of opposition
3722:Gödel's completeness
3521:The Theory of Groups
3337:
3317:
3297:
3277:
3251:
3247:of two vectors maps
3220:
3196:
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701:
651:
629:
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569:
512:
484:
457:
425:
397:
345:
322:
296:
269:
226:that are studied in
94:
74:
54:
34:
5304:Mathematical object
5195:P versus NP problem
5160:Computable function
4954:Reverse mathematics
4880:Logical consequence
4757:primitive recursive
4752:elementary function
4525:Free/bound variable
4378:TarskiâGrothendieck
3897:Logical connectives
3827:Logical equivalence
3677:Logical consequence
3007:may be viewed as a
2967:A binary operation
2799:{\displaystyle a+b}
2535:{\displaystyle a=0}
2301:{\displaystyle c=2}
2275:{\displaystyle b=3}
2249:{\displaystyle a=2}
230:, in particular in
30:A binary operation
5102:Transfer principle
5065:Semantics of logic
5050:Categorical theory
5026:Non-standard model
4540:Logical connective
3667:Information theory
3616:Mathematical logic
3571:"Binary Operation"
3568:Weisstein, Eric W.
3343:
3323:
3303:
3283:
3263:
3226:
3202:
3166:
3146:
3126:
3106:
3074:
3021:
2997:
2977:
2949:
2920:
2884:
2852:{\displaystyle ab}
2849:
2822:
2796:
2770:
2732:
2730:↑ ↑
2708:
2682:
2637:
2617:
2576:
2552:
2532:
2506:
2484:
2462:
2380:
2298:
2272:
2246:
2220:
2130:
2090:
2038:
2013:
1966:
1922:
1860:
1822:
1771:
1745:. Many also have
1735:
1715:
1695:
1675:
1655:
1565:
1545:
1525:
1505:
1441:
1421:
1401:
1381:
1361:
1334:
1307:
1281:
1144:
1106:
1074:
1054:
1031:
1005:
961:
935:
895:
869:
822:
796:
756:
709:
684:
637:
599:
575:
540:universal algebras
524:
490:
470:
431:
403:
373:
328:
308:
275:
117:
106:
80:
60:
40:
5353:Binary operations
5340:
5339:
5272:Abstract category
5075:Theories of truth
4885:Rule of inference
4875:Natural deduction
4856:
4855:
4401:
4400:
4106:Cartesian product
4011:
4010:
3917:Many-valued logic
3892:Boolean functions
3775:Russell's paradox
3750:diagonal argument
3647:First-order logic
3528:Hardy, Darel W.;
3483:978-0-387-77487-9
3472:Universal Algebra
3389:Ternary operation
3346:{\displaystyle K}
3326:{\displaystyle S}
3306:{\displaystyle K}
3286:{\displaystyle K}
3240:over that field.
3229:{\displaystyle S}
3205:{\displaystyle K}
3169:{\displaystyle S}
3149:{\displaystyle b}
3129:{\displaystyle a}
3024:{\displaystyle S}
3000:{\displaystyle S}
2980:{\displaystyle f}
2640:{\displaystyle a}
2579:{\displaystyle 1}
2555:{\displaystyle b}
1747:identity elements
1738:{\displaystyle S}
1718:{\displaystyle c}
1698:{\displaystyle b}
1678:{\displaystyle a}
1568:{\displaystyle S}
1548:{\displaystyle b}
1528:{\displaystyle a}
1515:for all elements
1444:{\displaystyle S}
1424:{\displaystyle C}
1404:{\displaystyle f}
1384:{\displaystyle S}
1077:{\displaystyle S}
1057:{\displaystyle C}
578:{\displaystyle +}
506:universal algebra
493:{\displaystyle a}
468:
434:{\displaystyle f}
406:{\displaystyle f}
331:{\displaystyle S}
291:Cartesian product
278:{\displaystyle S}
83:{\displaystyle y}
63:{\displaystyle x}
21:Bitwise operation
5360:
5331:
5330:
5282:History of logic
5277:Category of sets
5170:Decision problem
4949:Ordinal analysis
4890:Sequent calculus
4788:Boolean algebras
4728:
4727:
4702:
4673:logical/constant
4427:
4426:
4413:
4336:ZermeloâFraenkel
4087:Set operations:
4022:
4021:
3959:
3790:
3789:
3770:LöwenheimâSkolem
3657:Formal semantics
3609:
3602:
3595:
3586:
3585:
3581:
3580:
3555:
3546:
3530:Walker, Carol L.
3524:
3515:
3488:
3487:
3475:
3465:
3459:
3453:
3447:
3441:
3435:
3429:
3423:
3417:
3385:
3352:
3350:
3349:
3344:
3332:
3330:
3329:
3324:
3312:
3310:
3309:
3304:
3292:
3290:
3289:
3284:
3272:
3270:
3269:
3264:
3235:
3233:
3232:
3227:
3211:
3209:
3208:
3203:
3175:
3173:
3172:
3167:
3155:
3153:
3152:
3147:
3135:
3133:
3132:
3127:
3115:
3113:
3112:
3107:
3083:
3081:
3080:
3075:
3030:
3028:
3027:
3022:
3009:ternary relation
3006:
3004:
3003:
2998:
2986:
2984:
2983:
2978:
2958:
2956:
2955:
2950:
2929:
2927:
2926:
2921:
2893:
2891:
2890:
2885:
2858:
2856:
2855:
2850:
2836:with no symbol)
2831:
2829:
2828:
2823:
2805:
2803:
2802:
2797:
2779:
2777:
2776:
2771:
2741:
2739:
2738:
2733:
2717:
2715:
2714:
2709:
2691:
2689:
2688:
2683:
2646:
2644:
2643:
2638:
2626:
2624:
2623:
2618:
2585:
2583:
2582:
2577:
2561:
2559:
2558:
2553:
2541:
2539:
2538:
2533:
2515:
2513:
2512:
2507:
2505:
2493:
2491:
2490:
2485:
2483:
2471:
2469:
2468:
2463:
2455:
2454:
2418:
2417:
2389:
2387:
2386:
2381:
2373:
2372:
2330:
2329:
2307:
2305:
2304:
2299:
2281:
2279:
2278:
2273:
2255:
2253:
2252:
2247:
2229:
2227:
2226:
2221:
2139:
2137:
2136:
2131:
2129:
2128:
2116:
2115:
2099:
2097:
2096:
2091:
2089:
2088:
2047:
2045:
2044:
2039:
2037:
2022:
2020:
2019:
2014:
1975:
1973:
1972:
1967:
1932:; for instance,
1931:
1929:
1928:
1923:
1869:
1867:
1866:
1861:
1831:
1829:
1828:
1823:
1780:
1778:
1777:
1772:
1770:
1751:inverse elements
1744:
1742:
1741:
1736:
1724:
1722:
1721:
1716:
1704:
1702:
1701:
1696:
1684:
1682:
1681:
1676:
1664:
1662:
1661:
1656:
1574:
1572:
1571:
1566:
1554:
1552:
1551:
1546:
1534:
1532:
1531:
1526:
1514:
1512:
1511:
1506:
1450:
1448:
1447:
1442:
1430:
1428:
1427:
1422:
1410:
1408:
1407:
1402:
1390:
1388:
1387:
1382:
1370:
1368:
1367:
1362:
1360:
1359:
1343:
1341:
1340:
1335:
1333:
1332:
1316:
1314:
1313:
1308:
1290:
1288:
1287:
1282:
1268:
1267:
1255:
1254:
1230:
1229:
1217:
1216:
1189:
1188:
1176:
1175:
1153:
1151:
1150:
1145:
1115:
1113:
1112:
1107:
1083:
1081:
1080:
1075:
1063:
1061:
1060:
1055:
1044:For a given set
1040:
1038:
1037:
1032:
1014:
1012:
1011:
1006:
970:
968:
967:
962:
944:
942:
941:
936:
931:
904:
902:
901:
896:
878:
876:
875:
870:
831:
829:
828:
823:
805:
803:
802:
797:
792:
765:
763:
762:
757:
718:
716:
715:
710:
708:
693:
691:
690:
685:
646:
644:
643:
638:
636:
608:
606:
605:
600:
584:
582:
581:
576:
547:computer science
536:partial algebras
533:
531:
530:
525:
499:
497:
496:
491:
479:
477:
476:
471:
469:
461:
440:
438:
437:
432:
419:partial function
412:
410:
409:
404:
388:closure property
382:
380:
379:
374:
337:
335:
334:
329:
317:
315:
314:
309:
284:
282:
281:
276:
205:binary operation
152:binary operation
129:dyadic operation
125:binary operation
115:
113:
112:
107:
89:
87:
86:
81:
69:
67:
66:
61:
49:
47:
46:
41:
5368:
5367:
5363:
5362:
5361:
5359:
5358:
5357:
5343:
5342:
5341:
5336:
5325:
5318:
5263:Category theory
5253:Algebraic logic
5236:
5207:Lambda calculus
5145:Church encoding
5131:
5107:Truth predicate
4963:
4929:Complete theory
4852:
4721:
4717:
4713:
4708:
4700:
4420: and
4416:
4411:
4397:
4373:New Foundations
4341:axiom of choice
4324:
4286:Gödel numbering
4226: and
4218:
4122:
4007:
3957:
3938:
3887:Boolean algebra
3873:
3837:Equiconsistency
3802:Classical logic
3779:
3760:Halting problem
3748: and
3724: and
3712: and
3711:
3706:Theorems (
3701:
3618:
3613:
3562:
3544:
3513:
3497:
3492:
3491:
3484:
3466:
3462:
3454:
3450:
3442:
3438:
3430:
3426:
3418:
3414:
3409:
3400:Unary operation
3383:
3374:Magma (algebra)
3359:
3338:
3335:
3334:
3318:
3315:
3314:
3313:is a field and
3298:
3295:
3294:
3278:
3275:
3274:
3252:
3249:
3248:
3221:
3218:
3217:
3197:
3194:
3193:
3182:
3161:
3158:
3157:
3141:
3138:
3137:
3121:
3118:
3117:
3089:
3086:
3085:
3036:
3033:
3032:
3016:
3013:
3012:
2992:
2989:
2988:
2972:
2969:
2968:
2965:
2938:
2935:
2934:
2909:
2906:
2905:
2903:Polish notation
2864:
2861:
2860:
2841:
2838:
2837:
2811:
2808:
2807:
2785:
2782:
2781:
2759:
2756:
2755:
2748:
2727:
2724:
2723:
2703:
2700:
2699:
2656:
2653:
2652:
2632:
2629:
2628:
2591:
2588:
2587:
2571:
2568:
2567:
2547:
2544:
2543:
2521:
2518:
2517:
2501:
2499:
2496:
2495:
2479:
2477:
2474:
2473:
2450:
2446:
2413:
2409:
2395:
2392:
2391:
2368:
2364:
2325:
2321:
2313:
2310:
2309:
2287:
2284:
2283:
2261:
2258:
2257:
2235:
2232:
2231:
2149:
2146:
2145:
2124:
2120:
2111:
2107:
2105:
2102:
2101:
2084:
2080:
2057:
2054:
2053:
2033:
2031:
2028:
2027:
1981:
1978:
1977:
1937:
1934:
1933:
1875:
1872:
1871:
1837:
1834:
1833:
1790:
1787:
1786:
1766:
1764:
1761:
1760:
1730:
1727:
1726:
1710:
1707:
1706:
1690:
1687:
1686:
1670:
1667:
1666:
1584:
1581:
1580:
1560:
1557:
1556:
1540:
1537:
1536:
1520:
1517:
1516:
1464:
1461:
1460:
1436:
1433:
1432:
1416:
1413:
1412:
1396:
1393:
1392:
1376:
1373:
1372:
1355:
1351:
1349:
1346:
1345:
1328:
1324:
1322:
1319:
1318:
1296:
1293:
1292:
1263:
1259:
1250:
1246:
1225:
1221:
1212:
1208:
1184:
1180:
1171:
1167:
1159:
1156:
1155:
1121:
1118:
1117:
1089:
1086:
1085:
1069:
1066:
1065:
1049:
1046:
1045:
1020:
1017:
1016:
976:
973:
972:
950:
947:
946:
927:
913:
910:
909:
884:
881:
880:
837:
834:
833:
811:
808:
807:
788:
774:
771:
770:
724:
721:
720:
704:
702:
699:
698:
652:
649:
648:
632:
630:
627:
626:
594:
591:
590:
570:
567:
566:
559:
551:binary function
513:
510:
509:
485:
482:
481:
460:
458:
455:
454:
426:
423:
422:
398:
395:
394:
346:
343:
342:
323:
320:
319:
297:
294:
293:
270:
267:
266:
260:
207:. For example,
190:vector addition
162:binary function
95:
92:
91:
75:
72:
71:
55:
52:
51:
35:
32:
31:
24:
17:
12:
11:
5:
5366:
5356:
5355:
5338:
5337:
5323:
5320:
5319:
5317:
5316:
5311:
5306:
5301:
5296:
5295:
5294:
5284:
5279:
5274:
5265:
5260:
5255:
5250:
5248:Abstract logic
5244:
5242:
5238:
5237:
5235:
5234:
5229:
5227:Turing machine
5224:
5219:
5214:
5209:
5204:
5199:
5198:
5197:
5192:
5187:
5182:
5177:
5167:
5165:Computable set
5162:
5157:
5152:
5147:
5141:
5139:
5133:
5132:
5130:
5129:
5124:
5119:
5114:
5109:
5104:
5099:
5094:
5093:
5092:
5087:
5082:
5072:
5067:
5062:
5060:Satisfiability
5057:
5052:
5047:
5046:
5045:
5035:
5034:
5033:
5023:
5022:
5021:
5016:
5011:
5006:
5001:
4991:
4990:
4989:
4984:
4977:Interpretation
4973:
4971:
4965:
4964:
4962:
4961:
4956:
4951:
4946:
4941:
4931:
4926:
4925:
4924:
4923:
4922:
4912:
4907:
4897:
4892:
4887:
4882:
4877:
4872:
4866:
4864:
4858:
4857:
4854:
4853:
4851:
4850:
4842:
4841:
4840:
4839:
4834:
4833:
4832:
4827:
4822:
4802:
4801:
4800:
4798:minimal axioms
4795:
4784:
4783:
4782:
4771:
4770:
4769:
4764:
4759:
4754:
4749:
4744:
4731:
4729:
4710:
4709:
4707:
4706:
4705:
4704:
4692:
4687:
4686:
4685:
4680:
4675:
4670:
4660:
4655:
4650:
4645:
4644:
4643:
4638:
4628:
4627:
4626:
4621:
4616:
4611:
4601:
4596:
4595:
4594:
4589:
4584:
4574:
4573:
4572:
4567:
4562:
4557:
4552:
4547:
4537:
4532:
4527:
4522:
4521:
4520:
4515:
4510:
4505:
4495:
4490:
4488:Formation rule
4485:
4480:
4479:
4478:
4473:
4463:
4462:
4461:
4451:
4446:
4441:
4436:
4430:
4424:
4407:Formal systems
4403:
4402:
4399:
4398:
4396:
4395:
4390:
4385:
4380:
4375:
4370:
4365:
4360:
4355:
4350:
4349:
4348:
4343:
4332:
4330:
4326:
4325:
4323:
4322:
4321:
4320:
4310:
4305:
4304:
4303:
4296:Large cardinal
4293:
4288:
4283:
4278:
4273:
4259:
4258:
4257:
4252:
4247:
4232:
4230:
4220:
4219:
4217:
4216:
4215:
4214:
4209:
4204:
4194:
4189:
4184:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4133:
4131:
4124:
4123:
4121:
4120:
4119:
4118:
4113:
4108:
4103:
4098:
4093:
4085:
4084:
4083:
4078:
4068:
4063:
4061:Extensionality
4058:
4056:Ordinal number
4053:
4043:
4038:
4037:
4036:
4025:
4019:
4013:
4012:
4009:
4008:
4006:
4005:
4000:
3995:
3990:
3985:
3980:
3975:
3974:
3973:
3963:
3962:
3961:
3948:
3946:
3940:
3939:
3937:
3936:
3935:
3934:
3929:
3924:
3914:
3909:
3904:
3899:
3894:
3889:
3883:
3881:
3875:
3874:
3872:
3871:
3866:
3861:
3856:
3851:
3846:
3841:
3840:
3839:
3829:
3824:
3819:
3814:
3809:
3804:
3798:
3796:
3787:
3781:
3780:
3778:
3777:
3772:
3767:
3762:
3757:
3752:
3740:Cantor's
3738:
3733:
3728:
3718:
3716:
3703:
3702:
3700:
3699:
3694:
3689:
3684:
3679:
3674:
3669:
3664:
3659:
3654:
3649:
3644:
3639:
3638:
3637:
3626:
3624:
3620:
3619:
3612:
3611:
3604:
3597:
3589:
3583:
3582:
3561:
3560:External links
3558:
3557:
3556:
3547:
3542:
3525:
3516:
3511:
3496:
3493:
3490:
3489:
3482:
3460:
3448:
3436:
3424:
3411:
3410:
3408:
3405:
3404:
3403:
3397:
3392:
3386:
3377:
3371:
3365:
3358:
3355:
3342:
3322:
3302:
3282:
3262:
3259:
3256:
3225:
3201:
3190:linear algebra
3181:
3178:
3165:
3145:
3125:
3105:
3102:
3099:
3096:
3093:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3020:
2996:
2976:
2964:
2961:
2948:
2945:
2942:
2919:
2916:
2913:
2883:
2880:
2877:
2874:
2871:
2868:
2848:
2845:
2821:
2818:
2815:
2795:
2792:
2789:
2769:
2766:
2763:
2752:infix notation
2747:
2744:
2731:
2707:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2636:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2575:
2564:right identity
2551:
2531:
2528:
2525:
2504:
2482:
2461:
2458:
2453:
2449:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2416:
2412:
2408:
2405:
2402:
2399:
2379:
2376:
2371:
2367:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2328:
2324:
2320:
2317:
2297:
2294:
2291:
2271:
2268:
2265:
2245:
2242:
2239:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2142:Equation x = y
2127:
2123:
2119:
2114:
2110:
2087:
2083:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2050:exponentiation
2036:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1769:
1734:
1714:
1694:
1674:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1564:
1544:
1524:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1453:
1452:
1440:
1420:
1400:
1380:
1358:
1354:
1331:
1327:
1306:
1303:
1300:
1280:
1277:
1274:
1271:
1266:
1262:
1258:
1253:
1249:
1245:
1242:
1239:
1236:
1233:
1228:
1224:
1220:
1215:
1211:
1207:
1204:
1201:
1198:
1195:
1192:
1187:
1183:
1179:
1174:
1170:
1166:
1163:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1105:
1102:
1099:
1096:
1093:
1073:
1053:
1042:
1030:
1027:
1024:
1004:
1001:
998:
995:
992:
989:
986:
983:
980:
960:
957:
954:
934:
930:
926:
923:
920:
917:
906:
894:
891:
888:
868:
865:
862:
859:
856:
853:
850:
847:
844:
841:
821:
818:
815:
795:
791:
787:
784:
781:
778:
767:
755:
752:
749:
746:
743:
740:
737:
734:
731:
728:
707:
695:
683:
680:
677:
674:
671:
668:
665:
662:
659:
656:
635:
598:
587:multiplication
574:
558:
555:
523:
520:
517:
504:and classical
489:
467:
464:
451:divide by zero
430:
402:
384:
383:
372:
369:
366:
363:
360:
357:
354:
351:
327:
307:
304:
301:
274:
259:
256:
217:scalar product
186:multiplication
105:
102:
99:
79:
59:
39:
15:
9:
6:
4:
3:
2:
5365:
5354:
5351:
5350:
5348:
5335:
5334:
5329:
5321:
5315:
5312:
5310:
5307:
5305:
5302:
5300:
5297:
5293:
5290:
5289:
5288:
5285:
5283:
5280:
5278:
5275:
5273:
5269:
5266:
5264:
5261:
5259:
5256:
5254:
5251:
5249:
5246:
5245:
5243:
5239:
5233:
5230:
5228:
5225:
5223:
5222:Recursive set
5220:
5218:
5215:
5213:
5210:
5208:
5205:
5203:
5200:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5178:
5176:
5173:
5172:
5171:
5168:
5166:
5163:
5161:
5158:
5156:
5153:
5151:
5148:
5146:
5143:
5142:
5140:
5138:
5134:
5128:
5125:
5123:
5120:
5118:
5115:
5113:
5110:
5108:
5105:
5103:
5100:
5098:
5095:
5091:
5088:
5086:
5083:
5081:
5078:
5077:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5058:
5056:
5053:
5051:
5048:
5044:
5041:
5040:
5039:
5036:
5032:
5031:of arithmetic
5029:
5028:
5027:
5024:
5020:
5017:
5015:
5012:
5010:
5007:
5005:
5002:
5000:
4997:
4996:
4995:
4992:
4988:
4985:
4983:
4980:
4979:
4978:
4975:
4974:
4972:
4970:
4966:
4960:
4957:
4955:
4952:
4950:
4947:
4945:
4942:
4939:
4938:from ZFC
4935:
4932:
4930:
4927:
4921:
4918:
4917:
4916:
4913:
4911:
4908:
4906:
4903:
4902:
4901:
4898:
4896:
4893:
4891:
4888:
4886:
4883:
4881:
4878:
4876:
4873:
4871:
4868:
4867:
4865:
4863:
4859:
4849:
4848:
4844:
4843:
4838:
4837:non-Euclidean
4835:
4831:
4828:
4826:
4823:
4821:
4820:
4816:
4815:
4813:
4810:
4809:
4807:
4803:
4799:
4796:
4794:
4791:
4790:
4789:
4785:
4781:
4778:
4777:
4776:
4772:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4739:
4737:
4733:
4732:
4730:
4725:
4719:
4714:Example
4711:
4703:
4698:
4697:
4696:
4693:
4691:
4688:
4684:
4681:
4679:
4676:
4674:
4671:
4669:
4666:
4665:
4664:
4661:
4659:
4656:
4654:
4651:
4649:
4646:
4642:
4639:
4637:
4634:
4633:
4632:
4629:
4625:
4622:
4620:
4617:
4615:
4612:
4610:
4607:
4606:
4605:
4602:
4600:
4597:
4593:
4590:
4588:
4585:
4583:
4580:
4579:
4578:
4575:
4571:
4568:
4566:
4563:
4561:
4558:
4556:
4553:
4551:
4548:
4546:
4543:
4542:
4541:
4538:
4536:
4533:
4531:
4528:
4526:
4523:
4519:
4516:
4514:
4511:
4509:
4506:
4504:
4501:
4500:
4499:
4496:
4494:
4491:
4489:
4486:
4484:
4481:
4477:
4474:
4472:
4471:by definition
4469:
4468:
4467:
4464:
4460:
4457:
4456:
4455:
4452:
4450:
4447:
4445:
4442:
4440:
4437:
4435:
4432:
4431:
4428:
4425:
4423:
4419:
4414:
4408:
4404:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4369:
4366:
4364:
4361:
4359:
4358:KripkeâPlatek
4356:
4354:
4351:
4347:
4344:
4342:
4339:
4338:
4337:
4334:
4333:
4331:
4327:
4319:
4316:
4315:
4314:
4311:
4309:
4306:
4302:
4299:
4298:
4297:
4294:
4292:
4289:
4287:
4284:
4282:
4279:
4277:
4274:
4271:
4267:
4263:
4260:
4256:
4253:
4251:
4248:
4246:
4243:
4242:
4241:
4237:
4234:
4233:
4231:
4229:
4225:
4221:
4213:
4210:
4208:
4205:
4203:
4202:constructible
4200:
4199:
4198:
4195:
4193:
4190:
4188:
4185:
4183:
4180:
4178:
4175:
4173:
4170:
4168:
4165:
4163:
4160:
4158:
4155:
4153:
4150:
4148:
4145:
4143:
4140:
4138:
4135:
4134:
4132:
4130:
4125:
4117:
4114:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4092:
4089:
4088:
4086:
4082:
4079:
4077:
4074:
4073:
4072:
4069:
4067:
4064:
4062:
4059:
4057:
4054:
4052:
4048:
4044:
4042:
4039:
4035:
4032:
4031:
4030:
4027:
4026:
4023:
4020:
4018:
4014:
4004:
4001:
3999:
3996:
3994:
3991:
3989:
3986:
3984:
3981:
3979:
3976:
3972:
3969:
3968:
3967:
3964:
3960:
3955:
3954:
3953:
3950:
3949:
3947:
3945:
3941:
3933:
3930:
3928:
3925:
3923:
3920:
3919:
3918:
3915:
3913:
3910:
3908:
3905:
3903:
3900:
3898:
3895:
3893:
3890:
3888:
3885:
3884:
3882:
3880:
3879:Propositional
3876:
3870:
3867:
3865:
3862:
3860:
3857:
3855:
3852:
3850:
3847:
3845:
3842:
3838:
3835:
3834:
3833:
3830:
3828:
3825:
3823:
3820:
3818:
3815:
3813:
3810:
3808:
3807:Logical truth
3805:
3803:
3800:
3799:
3797:
3795:
3791:
3788:
3786:
3782:
3776:
3773:
3771:
3768:
3766:
3763:
3761:
3758:
3756:
3753:
3751:
3747:
3743:
3739:
3737:
3734:
3732:
3729:
3727:
3723:
3720:
3719:
3717:
3715:
3709:
3704:
3698:
3695:
3693:
3690:
3688:
3685:
3683:
3680:
3678:
3675:
3673:
3670:
3668:
3665:
3663:
3660:
3658:
3655:
3653:
3650:
3648:
3645:
3643:
3640:
3636:
3633:
3632:
3631:
3628:
3627:
3625:
3621:
3617:
3610:
3605:
3603:
3598:
3596:
3591:
3590:
3587:
3578:
3577:
3572:
3569:
3564:
3563:
3553:
3548:
3545:
3543:0-13-067464-8
3539:
3535:
3531:
3526:
3522:
3517:
3514:
3512:0-201-01984-1
3508:
3504:
3499:
3498:
3485:
3479:
3474:
3473:
3464:
3457:
3452:
3445:
3444:Fraleigh 1976
3440:
3433:
3428:
3421:
3416:
3412:
3401:
3398:
3396:
3393:
3390:
3387:
3381:
3378:
3375:
3372:
3369:
3366:
3364:
3361:
3360:
3354:
3340:
3320:
3300:
3280:
3260:
3257:
3254:
3246:
3241:
3239:
3223:
3215:
3199:
3191:
3187:
3184:For example,
3177:
3163:
3143:
3123:
3103:
3100:
3097:
3094:
3091:
3065:
3062:
3059:
3053:
3050:
3047:
3044:
3041:
3018:
3010:
2994:
2974:
2960:
2946:
2943:
2940:
2933:
2917:
2914:
2911:
2904:
2899:
2897:
2878:
2875:
2872:
2866:
2846:
2843:
2835:
2834:juxtaposition
2819:
2816:
2813:
2793:
2790:
2787:
2767:
2764:
2761:
2753:
2743:
2721:
2705:
2697:
2693:
2679:
2676:
2670:
2667:
2664:
2658:
2650:
2634:
2614:
2611:
2605:
2602:
2599:
2593:
2573:
2565:
2549:
2529:
2526:
2523:
2459:
2456:
2451:
2447:
2443:
2437:
2434:
2431:
2425:
2422:
2414:
2410:
2406:
2403:
2397:
2377:
2374:
2369:
2365:
2361:
2355:
2352:
2349:
2343:
2340:
2334:
2331:
2326:
2322:
2315:
2295:
2292:
2289:
2269:
2266:
2263:
2243:
2240:
2237:
2211:
2208:
2205:
2199:
2196:
2193:
2187:
2184:
2178:
2175:
2169:
2166:
2163:
2157:
2151:
2143:
2125:
2121:
2117:
2112:
2108:
2085:
2081:
2077:
2071:
2068:
2065:
2059:
2051:
2024:
2010:
2007:
2004:
2001:
1998:
1992:
1989:
1986:
1963:
1960:
1954:
1951:
1948:
1942:
1939:
1919:
1916:
1910:
1907:
1904:
1898:
1892:
1889:
1886:
1880:
1877:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1819:
1816:
1813:
1810:
1804:
1801:
1798:
1792:
1784:
1757:
1754:
1752:
1748:
1732:
1712:
1692:
1672:
1646:
1643:
1640:
1634:
1631:
1628:
1622:
1619:
1613:
1610:
1604:
1601:
1598:
1592:
1586:
1579:, satisfying
1578:
1562:
1542:
1522:
1499:
1496:
1493:
1487:
1484:
1478:
1475:
1472:
1466:
1459:, satisfying
1458:
1438:
1418:
1398:
1378:
1356:
1352:
1329:
1325:
1304:
1301:
1298:
1272:
1264:
1260:
1251:
1247:
1243:
1237:
1226:
1222:
1218:
1213:
1209:
1202:
1196:
1185:
1181:
1177:
1172:
1168:
1161:
1141:
1135:
1132:
1129:
1126:
1123:
1103:
1097:
1094:
1091:
1071:
1051:
1043:
1028:
1025:
1022:
1002:
999:
996:
990:
987:
984:
978:
958:
955:
952:
924:
921:
915:
907:
892:
889:
886:
866:
863:
860:
857:
851:
848:
845:
839:
819:
816:
813:
785:
782:
776:
768:
753:
750:
747:
744:
738:
735:
732:
726:
696:
681:
678:
675:
672:
666:
663:
660:
654:
624:
623:
622:
620:
616:
612:
596:
588:
572:
564:
554:
552:
548:
543:
541:
537:
521:
518:
515:
507:
503:
487:
465:
462:
452:
448:
444:
428:
420:
416:
400:
391:
389:
370:
367:
361:
358:
355:
352:
349:
341:
340:
339:
325:
305:
302:
299:
292:
288:
272:
265:
255:
253:
252:vector spaces
249:
245:
241:
237:
233:
229:
225:
220:
218:
214:
213:vector spaces
210:
206:
201:
199:
195:
191:
187:
183:
179:
175:
171:
167:
163:
159:
158:
153:
148:
146:
142:
138:
134:
130:
126:
122:
103:
100:
97:
77:
57:
37:
28:
22:
5324:
5122:Ultraproduct
4969:Model theory
4934:Independence
4870:Formal proof
4862:Proof theory
4845:
4818:
4775:real numbers
4747:second-order
4658:Substitution
4535:Metalanguage
4476:conservative
4449:Axiom schema
4393:Constructive
4363:MorseâKelley
4329:Set theories
4317:
4308:Aleph number
4301:inaccessible
4207:Grothendieck
4091:intersection
3978:Higher-order
3966:Second-order
3912:Truth tables
3869:Venn diagram
3652:Formal proof
3574:
3551:
3533:
3520:
3502:
3471:
3463:
3451:
3439:
3427:
3415:
3242:
3238:vector space
3183:
2966:
2900:
2749:
2694:
2692:in general.
2648:
2563:
2025:
1758:
1755:
1454:
560:
544:
502:model theory
447:real numbers
442:
441:is called a
392:
385:
261:
221:
204:
202:
154:
151:
149:
128:
124:
118:
5232:Type theory
5180:undecidable
5112:Truth value
4999:equivalence
4678:non-logical
4291:Enumeration
4281:Isomorphism
4228:cardinality
4212:Von Neumann
4177:Ultrafilter
4142:Uncountable
4076:equivalence
3993:Quantifiers
3983:Fixed-point
3952:First-order
3832:Consistency
3817:Proposition
3794:Traditional
3765:Lindström's
3755:Compactness
3697:Type theory
3642:Cardinality
3420:Rotman 1973
3245:dot product
2896:superscript
1785:, that is,
1783:subtraction
1577:associative
1457:commutative
908:On the set
769:On the set
617:as well as
538:generalize
534:. However,
500:. In both
258:Terminology
182:subtraction
121:mathematics
90:to produce
5043:elementary
4736:arithmetic
4604:Quantifier
4582:functional
4454:Expression
4172:Transitive
4116:identities
4101:complement
4034:hereditary
4017:Set theory
3495:References
2566:(which is
1116:. Define
232:semigroups
224:structures
164:whose two
5314:Supertask
5217:Recursion
5175:decidable
5009:saturated
4987:of models
4910:deductive
4905:axiomatic
4825:Hilbert's
4812:Euclidean
4793:canonical
4716:axiomatic
4648:Signature
4577:Predicate
4466:Extension
4388:Ackermann
4313:Operation
4192:Universal
4182:Recursive
4157:Singleton
4152:Inhabited
4137:Countable
4127:Types of
4111:power set
4081:partition
3998:Predicate
3944:Predicate
3859:Syllogism
3849:Soundness
3822:Inference
3812:Tautology
3714:paradoxes
3576:MathWorld
3456:Hall 1959
3258:×
3243:Also the
3101:×
3095:×
2987:on a set
2947:∗
2912:∗
2817:⋅
2765:∗
2720:Tetration
2706:÷
2677:≠
2185:≠
2118:≠
2008:−
1999:−
1990:−
1952:−
1943:−
1917:−
1908:−
1899:≠
1890:−
1881:−
1855:−
1849:≠
1843:−
1817:−
1302:∈
1219:∘
1139:→
1133:×
1127::
1101:→
1095::
1026:×
956:×
890:×
817:×
597:×
519:×
413:is not a
365:→
359:×
353::
303:×
141:operation
101:∘
38:∘
5347:Category
5299:Logicism
5292:timeline
5268:Concrete
5127:Validity
5097:T-schema
5090:Kripke's
5085:Tarski's
5080:semantic
5070:Strength
5019:submodel
5014:spectrum
4982:function
4830:Tarski's
4819:Elements
4806:geometry
4762:Robinson
4683:variable
4668:function
4641:spectrum
4631:Sentence
4587:variable
4530:Language
4483:Relation
4444:Automata
4434:Alphabet
4418:language
4272:-jection
4250:codomain
4236:Function
4197:Universe
4167:Infinite
4071:Relation
3854:Validity
3844:Argument
3742:theorem,
3532:(2002),
3446:, pg. 10
3357:See also
3293:, where
3192:. Here
3116:for all
2754:such as
2746:Notation
2696:Division
2649:identity
2627:for all
2586:) since
1665:for all
1291:for all
615:matrices
563:addition
415:function
178:addition
170:codomain
168:and the
137:operands
135:(called
133:elements
5241:Related
5038:Diagram
4936: (
4915:Hilbert
4900:Systems
4895:Theorem
4773:of the
4718:systems
4498:Formula
4493:Grammar
4409: (
4353:General
4066:Forcing
4051:Element
3971:Monadic
3746:paradox
3687:Theorem
3623:General
3458:, pg. 1
3422:, pg. 1
2832:or (by
1391:. Then
1041:matrix.
905:matrix.
611:numbers
421:, then
287:mapping
236:monoids
228:algebra
166:domains
5004:finite
4767:Skolem
4720:
4695:Theory
4663:Symbol
4653:String
4636:atomic
4513:ground
4508:closed
4503:atomic
4459:ground
4422:syntax
4318:binary
4245:domain
4162:Finite
3927:finite
3785:Logics
3744:
3692:Theory
3540:
3509:
3480:
2390:, but
2282:, and
1705:, and
1064:, let
585:) and
417:but a
250:, and
248:fields
240:groups
196:, and
184:, and
4994:Model
4742:Peano
4599:Proof
4439:Arity
4368:Naive
4255:image
4187:Fuzzy
4147:Empty
4096:union
4041:Class
3682:Model
3672:Lemma
3630:Axiom
3407:Notes
3236:is a
3214:field
3212:is a
2140:(cf.
1575:, or
609:) of
285:is a
244:rings
160:is a
155:on a
147:two.
145:arity
5117:Type
4920:list
4724:list
4701:list
4690:Term
4624:rank
4518:open
4412:list
4224:Maps
4129:sets
3988:Free
3958:list
3708:list
3635:list
3538:ISBN
3507:ISBN
3478:ISBN
3216:and
3136:and
2930:and
2542:and
1976:but
1749:and
1535:and
1344:and
613:and
386:The
123:, a
70:and
4804:of
4786:of
4734:of
4266:Sur
4240:Map
4047:Ur-
4029:Set
3273:to
3188:in
3156:in
3084:in
3011:on
2460:512
1725:in
1555:in
1371:in
1154:by
945:of
806:of
393:If
318:to
264:set
211:of
176:of
157:set
143:of
127:or
119:In
5349::
5190:NP
4814::
4808::
4738::
4415:),
4270:Bi
4262:In
3573:.
3176:.
2959:.
2898:.
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2308:,
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246:,
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5270:/
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2011:4
2005:=
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1993:2
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1955:3
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1946:(
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1914:)
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1500:a
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1479:b
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1467:f
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1357:2
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1270:(
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1238:c
1235:(
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1197:c
1194:(
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1182:h
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1165:(
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1000:A
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991:B
988:,
985:A
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929:R
925:,
922:2
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893:2
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864:+
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852:B
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846:A
843:(
840:f
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790:R
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739:b
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730:(
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706:N
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667:b
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