36:
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426:
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3238:
1510:{\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}{\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}{\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}
2454:
The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not
1586:. Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics.
3111:
2282:
2336:
517:
206:
of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for
323:
3337:
1246:
1136:
2134:
2020:
2188:
1970:
143:
2515:
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657:
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3001:
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701:
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1026:
982:
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415:
3357:
3233:{\displaystyle x\cdot {\mathrm {d} \over \mathrm {d} x}\psi =x\cdot \psi '\ \neq \ \psi +x\cdot \psi '={\mathrm {d} \over \mathrm {d} x}\left(x\cdot \psi \right)}
2781:
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2969:
2949:
2854:
2458:
Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function
2197:
374:
248:
2354:, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as
4104:
3389:, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.
2291:
264:
325:
In other words, an operation is commutative if every two elements commute. An operation that does not satisfy the above property is called
1141:
1031:
1975:
1925:
4147:
1825:
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3905:
2461:
3887:
3301:
2089:
2143:
2656:
3939:
3859:
3833:
3814:
3784:
3514:
2920:
178:, the property can also be used in more advanced settings. The name is needed because there are operations, such as
88:
2455:
change. In contrast, the commutative property states that the order of the terms does not affect the final result.
3254:, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear
3250:, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually
1801:
2914:
1787:
827:
for the functions are different when one changes the order of the operands. For example, the truth tables for
379:
3977:
3414:
1839:
35:
2870:
4172:
4167:
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demonstrate that commutativity is a property of particular connectives. The following are truth-functional
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20:
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1626:
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1594:
3251:
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That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative.
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1867:
1706:
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3044:
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2974:
2754:
2695:
2659:. Furthermore, associativity does not imply commutativity either – for example multiplication of
1871:
1712:
1699:
668:
3577:
2792:
710:
3794:
Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
2704:
1911:
1725:
1647:
987:
943:
618:
216:
179:
46:
3877:
3366:
3082:
429:
The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.
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3243:
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2664:
2380:
2355:
1820:
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1759:
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does not change the result. It is a fundamental property of many binary operations, and many
3500:
3342:
2660:
3004:
2449:
1732:
1581:
929:
800:. This property leads to two different "inverse" operations of exponentiation (namely, the
566:
2924:
2277:{\displaystyle {\big (}P\to (Q\to R){\big )}\leftrightarrow {\big (}Q\to (P\to R){\big )}}
8:
3429:
3409:
2760:
2750:
2431:
2409:
2073:
2069:
1915:
1907:
1832:
1815:
1777:
1738:
555:
1625:'s article entitled "On the real nature of symbolical algebra" published in 1840 in the
3534:
3281:
3261:
3003:. These two operators do not commute as may be seen by considering the effect of their
2954:
2934:
2860:
2839:
2746:
2698:
can be directly linked to commutativity. When a commutative operation is written as a
2420:
2395:
2065:
1745:
572:
359:
233:
212:
167:
3650:
2286:
Commutativity of equivalence (also called the complete commutative law of equivalence)
1564:
Records of the implicit use of the commutative property go back to ancient times. The
4162:
4004:
3935:
3912:
3883:
3855:
3829:
3810:
3780:
3531:
3510:
3424:
3247:
2077:
1858:
1851:
1663:
1654:
1640:
1529:
704:
613:
595:
580:
251:
3440:
Proof that Peano's axioms imply the commutativity of the addition of natural numbers
219:
is symmetric as two equal mathematical objects are equal regardless of their order.
2928:
2416:
2391:
1904:
228:
155:
67:
4030:
3803:
3360:
2699:
2046:
1808:
1601:
when describing functions that have what is now called the commutative property.
1272:
933:
543:
208:
3843:
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.
1580:
is known to have assumed the commutative property of multiplication in his book
2424:
2367:
2359:
1622:
1569:
820:
757:
551:
535:
527:
4035:
2370:
on real and complex numbers) is often used (or implicitly assumed) in proofs.
215:
if the relation applies regardless of the order of its operands; for example,
4141:
4007:
3077:
2401:
2384:
1751:
1525:
531:
2525:
does not affect the result), but it is not associative (since, for example,
3449:
2347:
2050:
1919:
1739:
1669:
1257:
937:
562:
1556:
3403:
1560:
The first known use of the term was in a French
Journal published in 1814
824:
662:
587:
547:
183:
151:
3906:"The Mathematical Legacy of Ancient Egypt – A Response To Robert Palter"
2331:{\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}
4039:
3989:
3419:
2686:
2351:
1733:
1249:
591:
425:
3869:
Abstract algebra theory. Uses commutativity property throughout book.
940:
to the real numbers is almost always noncommutative. For example, let
170:
depend on it. Perhaps most familiar as a property of arithmetic, e.g.
4012:
3985:
3923:
Article describing the mathematical ability of ancient civilizations.
3539:
2043:
1713:
808:
4081:
3255:
2363:
811:
operation), whereas multiplication only has one inverse operation.
801:
523:
203:
3996:
Definition of commutativity and examples of commutative operations
438:
3433:
2192:
Commutativity of implication (also called the law of permutation)
539:
163:
75:
56:
2908:
2863:
is analogous to a commutative operation, in that if a relation
1577:
1565:
2373:
607:
512:{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.}
3529:
2427:
is commutative. (Addition in a ring is always commutative.)
1802:
1760:
1726:
1681:
924:
3278:-direction of a particle are represented by the operators
3076:(also called products of operators) on a one-dimensional
1752:
3932:
1720:
318:{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}
3332:{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}
1833:
1788:
1694:
1670:
4128:
Biography of
Francois Servois, who first used the term
3047:
3012:
2977:
1476:
1437:
1401:
1362:
1326:
1287:
291:
4096:
Page covering the earliest uses of mathematical terms
3369:
3345:
3304:
3284:
3264:
3114:
3085:
2957:
2937:
2873:
2842:
2795:
2763:
2707:
2596:
2531:
2464:
2294:
2200:
2146:
2092:
2028:
1978:
1928:
1617:, meaning "to exchange" or "to switch", a cognate of
1281:
1260:
to itself (see below for the Matrix representation).
1241:{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}
1144:
1131:{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}
1034:
990:
946:
766:
713:
671:
627:
448:
382:
362:
267:
236:
91:
3363:). This is the same example except for the constant
2362:
the commutativity of well-known operations (such as
2129:{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}
2015:{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}
2690:
Graph showing the symmetry of the addition function
3852:Algebra: Abstract and Concrete, Stressing Symmetry
3802:
3381:
3351:
3331:
3290:
3270:
3232:
3100:
3068:
3033:
2995:
2963:
2943:
2897:
2848:
2828:
2775:
2737:
2667:is always associative but not always commutative.
2647:
2582:
2509:
2330:
2276:
2183:{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}
2182:
2128:
2034:
2014:
1965:{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}
1964:
1509:
1240:
1130:
1020:
976:
792:
746:
695:
651:
511:
409:
368:
317:
242:
137:
19:"Commutative" redirects here. For other uses, see
4102:
4052:
3713:
3711:
2434:both addition and multiplication are commutative.
1746:
4139:
4002:
1826:
1621:. The term then appeared in English in 1838. in
442:The addition of vectors is commutative, because
4046:Examples proving some noncommutative operations
3494:
2056:
1795:
1687:
198:. The idea that simple operations, such as the
4073:Article giving the history of the real numbers
3708:
3655:Transactions of the Royal Society of Edinburgh
1627:Transactions of the Royal Society of Edinburgh
1275:is almost always noncommutative, for example:
703:. However it is classified more precisely as
138:{\displaystyle x*y=y*x\quad \forall x,y\in S.}
3930:Gay, Robins R.; Shute, Charles C. D. (1987).
2269:
2241:
2231:
2203:
1840:
1809:
1707:
1700:
1605:is the feminine form of the French adjective
3875:
3691:
2909:Non-commuting operators in quantum mechanics
2517:which is clearly commutative (interchanging
602:
4082:"Earliest Known Uses of Mathematical Terms"
222:
4079:
3651:"On the real nature of symbolical algebra"
3548:
3476:
3472:
3470:
34:
3876:Hurley, Patrick J.; Watson, Lori (2016).
3800:
3680:
3490:
3488:
2374:Mathematical structures and commutativity
1895:In truth-functional propositional logic,
608:Division, subtraction, and exponentiation
433:
3929:
3911:(Unpublished manuscript). Archived from
3613:
2927:, physical variables are represented by
2685:
2510:{\displaystyle f(x,y)={\frac {x+y}{2}},}
1794:
1609:, which is derived from the French noun
1555:
1551:
1528:) of two vectors in three dimensions is
1263:
925:Function composition of linear functions
437:
424:
16:Property of some mathematical operations
3903:
3849:
3823:
3753:
3741:
3729:
3717:
3648:
3601:
3467:
194:commutative, and so are referred to as
4140:
4036:Examples of non-commutative operations
3949:Translation and interpretation of the
3485:
2898:{\displaystyle aRb\Leftrightarrow bRa}
2655:). More such examples may be found in
1890:
1632:
4003:
3801:Copi, Irving M.; Cohen, Carl (2005).
3774:
3702:
3530:
2438:
2412:whose group operation is commutative.
2390:If the operation additionally has an
1248:This also applies more generally for
186:, that do not have it (for example,
3960:
3882:(12th ed.). Cengage Learning.
2049:representing "can be replaced in a
1910:. The rules allow one to transpose
1589:The first recorded use of the term
652:{\displaystyle 1\div 2\neq 2\div 1}
13:
3320:
3316:
3201:
3195:
3130:
3124:
2657:commutative non-associative magmas
814:
211:; a binary relation is said to be
114:
14:
4184:
3828:(6e ed.). Houghton Mifflin.
3376:
3311:
1568:used the commutative property of
1519:
561:Addition is commutative in every
4103:O'Conner, J.J.; Robertson, E.F.
4053:O'Conner, J.J.; Robertson, E.F.
4029:
3809:(12th ed.). Prentice Hall.
3502:Mathematics in Victorian Britain
2648:{\displaystyle f(f(-4,0),+4)=+1}
2583:{\displaystyle f(-4,f(0,+4))=-1}
2443:
2035:{\displaystyle \Leftrightarrow }
4148:Properties of binary operations
4105:"biography of François Servois"
4080:Cabillón, Julio; Miller, Jeff.
4023:Explanation of the term commute
3879:A Concise Introduction to Logic
3747:
3735:
3723:
3696:
3685:
3674:
3665:
3642:
3630:
2783:. For example, if the function
2745:then this function is called a
2670:
2383:is a set endowed with a total,
793:{\displaystyle 2^{3}\neq 3^{2}}
289:
113:
3854:(2e ed.). Prentice Hall.
3618:
3607:
3595:
3570:
3557:
3523:
3495:Flood, Raymond; Rice, Adrian;
3095:
3089:
2915:Canonical commutation relation
2883:
2811:
2799:
2757:is symmetric across the plane
2729:
2717:
2633:
2621:
2606:
2600:
2568:
2565:
2550:
2535:
2480:
2468:
2325:
2319:
2313:
2310:
2307:
2301:
2295:
2264:
2258:
2252:
2249:
2236:
2226:
2220:
2214:
2211:
2177:
2165:
2162:
2159:
2147:
2123:
2111:
2108:
2105:
2093:
2029:
2009:
1997:
1994:
1991:
1979:
1959:
1947:
1944:
1941:
1929:
1214:
1199:
1190:
1187:
1181:
1175:
1166:
1160:
1157:
1145:
1104:
1089:
1080:
1077:
1071:
1065:
1056:
1050:
1047:
1035:
1000:
994:
956:
950:
823:are noncommutative, since the
741:
729:
579:are commutative operations on
534:, and, in particular, between
500:
485:
470:
455:
1:
3826:Contemporary Abstract Algebra
3777:Linear Algebra Done Right, 2e
3763:
3415:Commutative (neurophysiology)
3069:{\textstyle {\frac {d}{dx}}x}
3034:{\textstyle x{\frac {d}{dx}}}
2341:
1597:in 1814, which used the word
554:. This is also true in every
162:if changing the order of the
74:if changing the order of the
3481:Commutative and Distributive
2996:{\textstyle {\frac {d}{dx}}}
2138:Commutativity of disjunction
2084:Commutativity of conjunction
2057:Truth functional connectives
21:Commutative (disambiguation)
7:
3992:., Accessed 8 August 2007.
3973:Encyclopedia of Mathematics
3897:
3850:Goodman, Frederick (2003).
3392:
2681:
696:{\displaystyle 0-1\neq 1-0}
420:
78:does not change the result.
10:
4189:
4042:., Accessed 8 August 2007
4019:, Accessed 8 August 2007.
3951:Rhind Mathematical Papyrus
3636:O'Conner & Robertson,
3582:Mathematics Stack Exchange
3445:Quasi-commutative property
3404:Centralizer and normalizer
2912:
2829:{\displaystyle f(x,y)=x+y}
2674:
2447:
2387:and commutative operation.
1877:Existential generalization
1682:Biconditional introduction
747:{\displaystyle 0-1=-(1-0)}
611:
18:
4055:"History of real numbers"
3624:O'Conner & Robertson
3578:"User MathematicalOrchid"
3406:(also called a commutant)
2856:is a symmetric function.
2738:{\displaystyle z=f(x,y),}
1021:{\displaystyle g(x)=3x+7}
977:{\displaystyle f(x)=2x+1}
857:
852:
847:
842:
760:is noncommutative, since
665:is noncommutative, since
621:is noncommutative, since
603:Noncommutative operations
196:noncommutative operations
82:
62:
52:
42:
33:
3824:Gallian, Joseph (2006).
3768:
3692:Hurley & Watson 2016
3460:
3399:Anticommutative property
3382:{\displaystyle -i\hbar }
3101:{\displaystyle \psi (x)}
1868:Universal generalization
1708:Disjunction introduction
1695:Conjunction introduction
1665:Implication introduction
530:are commutative in most
410:{\displaystyle x*y=y*x.}
223:Mathematical definitions
3775:Axler, Sheldon (1997).
3649:Gregory, D. F. (1840).
3507:Oxford University Press
3361:reduced Planck constant
2755:three-dimensional space
1912:propositional variables
1524:The vector product (or
190:); such operations are
3432:(for commutativity in
3383:
3353:
3352:{\displaystyle \hbar }
3339:, respectively (where
3333:
3292:
3272:
3234:
3102:
3070:
3035:
2997:
2965:
2945:
2899:
2850:
2830:
2777:
2739:
2691:
2649:
2584:
2511:
2332:
2278:
2184:
2130:
2064:is a property of some
2036:
2016:
1966:
1727:hypothetical syllogism
1648:Propositional calculus
1572:to simplify computing
1561:
1511:
1254:affine transformations
1242:
1132:
1022:
978:
794:
748:
697:
653:
519:
513:
434:Commutative operations
430:
411:
370:
319:
244:
139:
3805:Introduction to Logic
3681:Copi & Cohen 2005
3477:Cabillón & Miller
3455:Commuting probability
3384:
3354:
3334:
3293:
3273:
3244:uncertainty principle
3235:
3103:
3071:
3036:
2998:
2966:
2951:(meaning multiply by
2946:
2900:
2851:
2831:
2778:
2740:
2689:
2677:Distributive property
2650:
2585:
2512:
2381:commutative semigroup
2333:
2279:
2185:
2131:
2037:
2017:
1967:
1769:Negation introduction
1762:modus ponendo tollens
1559:
1552:History and etymology
1512:
1269:Matrix multiplication
1264:Matrix multiplication
1243:
1133:
1023:
979:
795:
749:
698:
654:
514:
441:
428:
412:
371:
320:
245:
140:
3904:Lumpkin, B. (1997).
3614:Gay & Shute 1987
3535:"Symmetric Relation"
3367:
3343:
3302:
3282:
3262:
3112:
3083:
3045:
3010:
2975:
2955:
2935:
2871:
2840:
2793:
2761:
2705:
2594:
2529:
2462:
2450:Associative property
2292:
2198:
2144:
2090:
2074:logical equivalences
2068:of truth functional
2026:
1976:
1926:
1908:rules of replacement
1827:Material implication
1778:Rules of replacement
1641:Transformation rules
1613:and the French verb
1279:
1142:
1032:
988:
944:
930:Function composition
764:
711:
669:
625:
446:
380:
360:
265:
234:
89:
29:Commutative property
4173:Functional analysis
4168:Concepts in physics
4115:on 2 September 2009
3430:Particle statistics
3410:Commutative diagram
2867:is symmetric, then
2776:{\displaystyle y=x}
2070:propositional logic
2066:logical connectives
1916:logical expressions
1891:Rule of replacement
1740:destructive dilemma
1633:Propositional logic
1593:was in a memoir by
168:mathematical proofs
30:
4158:Rules of inference
4153:Elementary algebra
4005:Weisstein, Eric W.
3934:. British Museum.
3532:Weisstein, Eric W.
3379:
3349:
3329:
3288:
3268:
3230:
3098:
3066:
3031:
2993:
2961:
2941:
2895:
2861:symmetric relation
2846:
2826:
2773:
2747:symmetric function
2735:
2692:
2645:
2580:
2507:
2439:Related properties
2396:commutative monoid
2328:
2274:
2180:
2126:
2032:
2012:
1962:
1859:Rules of inference
1655:Rules of inference
1562:
1507:
1501:
1462:
1426:
1387:
1351:
1312:
1238:
1128:
1018:
974:
833:(B ⇒ A) = (A ∨ ¬B)
829:(A ⇒ B) = (¬A ∨ B)
807:operation and the
790:
744:
693:
649:
596:logical operations
594:" are commutative
520:
509:
431:
407:
366:
315:
295:
240:
135:
83:Symbolic statement
28:
3889:978-1-337-51478-1
3732:, pp. 26, 87
3425:Parallelogram law
3327:
3291:{\displaystyle x}
3271:{\displaystyle x}
3242:According to the
3209:
3168:
3162:
3138:
3061:
3029:
2991:
2964:{\displaystyle x}
2944:{\displaystyle x}
2923:as formulated by
2921:quantum mechanics
2859:For relations, a
2849:{\displaystyle f}
2502:
2406:commutative group
1922:. The rules are:
1888:
1887:
920:
919:
503:
488:
473:
458:
369:{\displaystyle *}
294:
243:{\displaystyle *}
148:
147:
4180:
4124:
4122:
4120:
4111:. Archived from
4092:
4090:
4088:
4069:
4067:
4065:
4034:
4018:
4017:
3981:
3961:Online resources
3945:
3919:
3918:on 13 July 2007.
3917:
3910:
3893:
3865:
3839:
3820:
3808:
3790:
3757:
3751:
3745:
3739:
3733:
3727:
3721:
3715:
3706:
3700:
3694:
3689:
3683:
3678:
3672:
3671:Moore and Parker
3669:
3663:
3662:
3646:
3640:
3634:
3628:
3622:
3616:
3611:
3605:
3599:
3593:
3592:
3590:
3588:
3574:
3568:
3561:
3555:
3552:
3546:
3545:
3544:
3527:
3521:
3520:
3492:
3483:
3474:
3388:
3386:
3385:
3380:
3358:
3356:
3355:
3350:
3338:
3336:
3335:
3330:
3328:
3326:
3315:
3297:
3295:
3294:
3289:
3277:
3275:
3274:
3269:
3239:
3237:
3236:
3231:
3229:
3225:
3210:
3208:
3204:
3198:
3193:
3188:
3166:
3160:
3159:
3139:
3137:
3133:
3127:
3122:
3107:
3105:
3104:
3099:
3075:
3073:
3072:
3067:
3062:
3060:
3049:
3040:
3038:
3037:
3032:
3030:
3028:
3017:
3002:
3000:
2999:
2994:
2992:
2990:
2979:
2970:
2968:
2967:
2962:
2950:
2948:
2947:
2942:
2929:linear operators
2904:
2902:
2901:
2896:
2855:
2853:
2852:
2847:
2835:
2833:
2832:
2827:
2788:
2782:
2780:
2779:
2774:
2744:
2742:
2741:
2736:
2654:
2652:
2651:
2646:
2589:
2587:
2586:
2581:
2516:
2514:
2513:
2508:
2503:
2498:
2487:
2417:commutative ring
2392:identity element
2337:
2335:
2334:
2329:
2283:
2281:
2280:
2275:
2273:
2272:
2245:
2244:
2235:
2234:
2207:
2206:
2189:
2187:
2186:
2181:
2135:
2133:
2132:
2127:
2072:. The following
2041:
2039:
2038:
2033:
2021:
2019:
2018:
2013:
1971:
1969:
1968:
1963:
1842:
1835:
1828:
1816:De Morgan's laws
1811:
1804:
1797:
1790:
1764:
1756:
1748:
1741:
1735:
1728:
1722:
1715:
1709:
1702:
1696:
1689:
1683:
1676:
1666:
1637:
1636:
1595:François Servois
1530:anti-commutative
1516:
1514:
1513:
1508:
1506:
1505:
1467:
1466:
1431:
1430:
1392:
1391:
1356:
1355:
1317:
1316:
1247:
1245:
1244:
1239:
1137:
1135:
1134:
1129:
1027:
1025:
1024:
1019:
983:
981:
980:
975:
934:linear functions
860:
855:
850:
845:
840:
839:
834:
830:
799:
797:
796:
791:
789:
788:
776:
775:
753:
751:
750:
745:
705:anti-commutative
702:
700:
699:
694:
658:
656:
655:
650:
614:Equation xy = yx
544:rational numbers
518:
516:
515:
510:
505:
504:
496:
490:
489:
481:
475:
474:
466:
460:
459:
451:
416:
414:
413:
408:
375:
373:
372:
367:
352:
348:
344:
335:
324:
322:
321:
316:
296:
292:
249:
247:
246:
241:
229:binary operation
209:binary relations
189:
177:
173:
156:binary operation
144:
142:
141:
136:
68:binary operation
38:
31:
27:
4188:
4187:
4183:
4182:
4181:
4179:
4178:
4177:
4138:
4137:
4135:
4118:
4116:
4086:
4084:
4063:
4061:
3984:Krowne, Aaron,
3968:"Commutativity"
3966:
3963:
3942:
3915:
3908:
3900:
3890:
3862:
3836:
3817:
3787:
3771:
3766:
3761:
3760:
3752:
3748:
3740:
3736:
3728:
3724:
3716:
3709:
3701:
3697:
3690:
3686:
3679:
3675:
3670:
3666:
3647:
3643:
3635:
3631:
3623:
3619:
3612:
3608:
3600:
3596:
3586:
3584:
3576:
3575:
3571:
3562:
3558:
3553:
3549:
3528:
3524:
3517:
3499:, eds. (2011).
3493:
3486:
3475:
3468:
3463:
3395:
3368:
3365:
3364:
3344:
3341:
3340:
3319:
3314:
3303:
3300:
3299:
3283:
3280:
3279:
3263:
3260:
3259:
3215:
3211:
3200:
3199:
3194:
3192:
3181:
3152:
3129:
3128:
3123:
3121:
3113:
3110:
3109:
3084:
3081:
3080:
3053:
3048:
3046:
3043:
3042:
3021:
3016:
3011:
3008:
3007:
2983:
2978:
2976:
2973:
2972:
2956:
2953:
2952:
2936:
2933:
2932:
2917:
2911:
2872:
2869:
2868:
2841:
2838:
2837:
2794:
2791:
2790:
2784:
2762:
2759:
2758:
2706:
2703:
2702:
2700:binary function
2684:
2679:
2673:
2595:
2592:
2591:
2530:
2527:
2526:
2488:
2486:
2463:
2460:
2459:
2452:
2446:
2441:
2376:
2344:
2293:
2290:
2289:
2268:
2267:
2240:
2239:
2230:
2229:
2202:
2201:
2199:
2196:
2195:
2145:
2142:
2141:
2091:
2088:
2087:
2059:
2027:
2024:
2023:
1977:
1974:
1973:
1927:
1924:
1923:
1893:
1852:Predicate logic
1846:
1810:Double negation
1664:
1635:
1554:
1522:
1500:
1499:
1494:
1488:
1487:
1482:
1472:
1471:
1461:
1460:
1455:
1449:
1448:
1443:
1433:
1432:
1425:
1424:
1419:
1413:
1412:
1407:
1397:
1396:
1386:
1385:
1380:
1374:
1373:
1368:
1358:
1357:
1350:
1349:
1344:
1338:
1337:
1332:
1322:
1321:
1311:
1310:
1305:
1299:
1298:
1293:
1283:
1282:
1280:
1277:
1276:
1273:square matrices
1266:
1143:
1140:
1139:
1033:
1030:
1029:
989:
986:
985:
945:
942:
941:
927:
858:
853:
848:
843:
832:
828:
821:truth functions
817:
815:Truth functions
784:
780:
771:
767:
765:
762:
761:
712:
709:
708:
670:
667:
666:
626:
623:
622:
616:
610:
605:
552:complex numbers
536:natural numbers
495:
494:
480:
479:
465:
464:
450:
449:
447:
444:
443:
436:
423:
381:
378:
377:
361:
358:
357:
350:
346:
340:
333:
290:
266:
263:
262:
235:
232:
231:
225:
188:"3 − 5 ≠ 5 − 3"
187:
176:"2 × 5 = 5 × 2"
175:
172:"3 + 4 = 4 + 3"
171:
90:
87:
86:
24:
17:
12:
11:
5:
4186:
4176:
4175:
4170:
4165:
4160:
4155:
4150:
4133:
4132:
4131:
4130:
4100:
4099:
4098:
4077:
4076:
4075:
4050:
4049:
4048:
4027:
4026:
4025:
4000:
3999:
3998:
3982:
3962:
3959:
3958:
3957:
3956:
3955:
3940:
3927:
3926:
3925:
3899:
3896:
3895:
3894:
3888:
3873:
3872:
3871:
3860:
3847:
3846:
3845:
3834:
3821:
3815:
3798:
3797:
3796:
3785:
3770:
3767:
3765:
3762:
3759:
3758:
3746:
3734:
3722:
3707:
3695:
3684:
3673:
3664:
3641:
3629:
3617:
3606:
3594:
3569:
3556:
3547:
3522:
3515:
3484:
3465:
3464:
3462:
3459:
3458:
3457:
3452:
3447:
3442:
3437:
3427:
3422:
3417:
3412:
3407:
3401:
3394:
3391:
3378:
3375:
3372:
3348:
3325:
3322:
3318:
3313:
3310:
3307:
3287:
3267:
3228:
3224:
3221:
3218:
3214:
3207:
3203:
3197:
3191:
3187:
3184:
3180:
3177:
3174:
3171:
3165:
3158:
3155:
3151:
3148:
3145:
3142:
3136:
3132:
3126:
3120:
3117:
3097:
3094:
3091:
3088:
3065:
3059:
3056:
3052:
3027:
3024:
3020:
3015:
2989:
2986:
2982:
2960:
2940:
2913:Main article:
2910:
2907:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2845:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2789:is defined as
2772:
2769:
2766:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2694:Some forms of
2683:
2680:
2675:Main article:
2672:
2669:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2579:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2534:
2506:
2501:
2497:
2494:
2491:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2448:Main article:
2445:
2442:
2440:
2437:
2436:
2435:
2428:
2425:multiplication
2413:
2398:
2388:
2375:
2372:
2368:multiplication
2360:linear algebra
2343:
2340:
2339:
2338:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2287:
2284:
2271:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2243:
2238:
2233:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2205:
2193:
2190:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2139:
2136:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2095:
2085:
2058:
2055:
2031:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1920:logical proofs
1892:
1889:
1886:
1885:
1884:
1883:
1874:
1862:
1861:
1855:
1854:
1848:
1847:
1845:
1844:
1837:
1830:
1823:
1818:
1813:
1806:
1803:Distributivity
1799:
1792:
1784:
1781:
1780:
1774:
1773:
1772:
1771:
1766:
1743:
1730:
1717:
1704:
1691:
1678:
1658:
1657:
1651:
1650:
1644:
1643:
1634:
1631:
1623:Duncan Gregory
1570:multiplication
1553:
1550:
1521:
1520:Vector product
1518:
1504:
1498:
1495:
1493:
1490:
1489:
1486:
1483:
1481:
1478:
1477:
1475:
1470:
1465:
1459:
1456:
1454:
1451:
1450:
1447:
1444:
1442:
1439:
1438:
1436:
1429:
1423:
1420:
1418:
1415:
1414:
1411:
1408:
1406:
1403:
1402:
1400:
1395:
1390:
1384:
1381:
1379:
1376:
1375:
1372:
1369:
1367:
1364:
1363:
1361:
1354:
1348:
1345:
1343:
1340:
1339:
1336:
1333:
1331:
1328:
1327:
1325:
1320:
1315:
1309:
1306:
1304:
1301:
1300:
1297:
1294:
1292:
1289:
1288:
1286:
1265:
1262:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1127:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
973:
970:
967:
964:
961:
958:
955:
952:
949:
926:
923:
922:
921:
918:
917:
914:
911:
908:
904:
903:
900:
897:
894:
890:
889:
886:
883:
880:
876:
875:
872:
869:
866:
862:
861:
856:
851:
846:
816:
813:
787:
783:
779:
774:
770:
758:Exponentiation
743:
740:
737:
734:
731:
728:
725:
722:
719:
716:
692:
689:
686:
683:
680:
677:
674:
648:
645:
642:
639:
636:
633:
630:
609:
606:
604:
601:
600:
599:
584:
570:
559:
532:number systems
528:multiplication
508:
502:
499:
493:
487:
484:
478:
472:
469:
463:
457:
454:
435:
432:
422:
419:
406:
403:
400:
397:
394:
391:
388:
385:
365:
332:One says that
327:noncommutative
314:
311:
308:
305:
302:
299:
288:
285:
282:
279:
276:
273:
270:
239:
224:
221:
200:multiplication
146:
145:
134:
131:
128:
125:
122:
119:
116:
112:
109:
106:
103:
100:
97:
94:
84:
80:
79:
64:
60:
59:
54:
50:
49:
44:
40:
39:
15:
9:
6:
4:
3:
2:
4185:
4174:
4171:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4149:
4146:
4145:
4143:
4136:
4129:
4126:
4125:
4114:
4110:
4106:
4101:
4097:
4094:
4093:
4083:
4078:
4074:
4071:
4070:
4060:
4056:
4051:
4047:
4044:
4043:
4041:
4037:
4032:
4028:
4024:
4021:
4020:
4015:
4014:
4009:
4006:
4001:
3997:
3994:
3993:
3991:
3987:
3983:
3979:
3975:
3974:
3969:
3965:
3964:
3954:
3952:
3947:
3946:
3943:
3941:0-7141-0944-4
3937:
3933:
3928:
3924:
3921:
3920:
3914:
3907:
3902:
3901:
3891:
3885:
3881:
3880:
3874:
3870:
3867:
3866:
3863:
3861:0-13-067342-0
3857:
3853:
3848:
3844:
3841:
3840:
3837:
3835:0-618-51471-6
3831:
3827:
3822:
3818:
3816:9780131898349
3812:
3807:
3806:
3799:
3795:
3792:
3791:
3788:
3786:0-387-98258-2
3782:
3778:
3773:
3772:
3756:, p. 250
3755:
3750:
3744:, p. 236
3743:
3738:
3731:
3726:
3719:
3714:
3712:
3704:
3699:
3693:
3688:
3682:
3677:
3668:
3660:
3656:
3652:
3645:
3639:
3633:
3627:
3621:
3615:
3610:
3603:
3598:
3583:
3579:
3573:
3566:
3560:
3551:
3542:
3541:
3536:
3533:
3526:
3518:
3516:9780191627941
3512:
3509:. p. 4.
3508:
3504:
3503:
3498:
3497:Wilson, Robin
3491:
3489:
3482:
3478:
3473:
3471:
3466:
3456:
3453:
3451:
3448:
3446:
3443:
3441:
3438:
3435:
3431:
3428:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3405:
3402:
3400:
3397:
3396:
3390:
3373:
3370:
3362:
3346:
3323:
3308:
3305:
3285:
3265:
3257:
3253:
3252:complementary
3249:
3245:
3240:
3226:
3222:
3219:
3216:
3212:
3205:
3189:
3185:
3182:
3178:
3175:
3172:
3169:
3163:
3156:
3153:
3149:
3146:
3143:
3140:
3134:
3118:
3115:
3092:
3086:
3079:
3078:wave function
3063:
3057:
3054:
3050:
3025:
3022:
3018:
3013:
3006:
2987:
2984:
2980:
2958:
2938:
2930:
2926:
2922:
2916:
2906:
2892:
2889:
2886:
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2492:
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2477:
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2465:
2456:
2451:
2444:Associativity
2433:
2429:
2426:
2422:
2418:
2414:
2411:
2407:
2403:
2402:abelian group
2399:
2397:
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2062:Commutativity
2054:
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2006:
2003:
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1985:
1982:
1956:
1953:
1950:
1938:
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1902:
1901:commutativity
1898:
1882:
1881:instantiation
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1873:
1872:instantiation
1869:
1866:
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1821:Transposition
1819:
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1800:
1798:
1796:Commutativity
1793:
1791:
1789:Associativity
1786:
1785:
1783:
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1767:
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1754:modus tollens
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565:and in every
564:
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293:for all
286:
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4113:the original
4108:
4095:
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4072:
4062:. Retrieved
4058:
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3971:
3948:
3931:
3922:
3913:the original
3878:
3868:
3851:
3842:
3825:
3804:
3793:
3779:. Springer.
3776:
3754:Gallian 2006
3749:
3742:Gallian 2006
3737:
3730:Gallian 2006
3725:
3720:, p. 34
3718:Gallian 2006
3698:
3687:
3676:
3667:
3658:
3654:
3644:
3637:
3632:
3626:Real Numbers
3625:
3620:
3609:
3604:, p. 11
3602:Lumpkin 1997
3597:
3585:. Retrieved
3581:
3572:
3564:
3559:
3554:Krowne, p. 1
3550:
3538:
3525:
3501:
3480:
3450:Trace monoid
3241:
3005:compositions
2918:
2864:
2858:
2785:
2693:
2671:Distributive
2522:
2518:
2457:
2453:
2405:
2394:, we have a
2345:
2061:
2060:
1900:
1896:
1894:
1879: /
1870: /
1761:
1758: /
1753:
1750: /
1737: /
1734:Constructive
1724: /
1711: /
1698: /
1685: /
1673:modus ponens
1672:
1668: /
1618:
1614:
1610:
1606:
1602:
1599:commutatives
1598:
1590:
1588:
1582:
1563:
1545:
1541:
1537:
1533:
1523:
1267:
1258:vector space
938:real numbers
928:
825:truth tables
818:
802:
756:
661:
617:
577:intersection
563:vector space
548:real numbers
353:
341:
336:
331:
326:
258:
254:
226:
195:
191:
159:
149:
71:
25:
4087:22 November
3986:Commutative
3705:, p. 2
3563:Weisstein,
2925:Schrödinger
2661:quaternions
2385:associative
2078:tautologies
2044:metalogical
1897:commutation
1834:Exportation
1721:Disjunctive
1714:elimination
1701:elimination
1688:elimination
1611:commutation
1603:Commutative
1591:commutative
663:Subtraction
259:commutative
184:subtraction
160:commutative
152:mathematics
72:commutative
4142:Categories
4040:PlanetMath
3990:PlanetMath
3764:References
3703:Axler 1997
3661:: 208–216.
3587:20 January
3420:Commutator
3248:Heisenberg
2749:, and its
2352:set theory
2342:Set theory
1747:Absorption
1619:to commute
1607:commutatif
612:See also:
257:is called
4013:MathWorld
4008:"Commute"
3978:EMS Press
3540:MathWorld
3377:ℏ
3371:−
3347:ℏ
3321:∂
3317:∂
3312:ℏ
3306:−
3223:ψ
3220:⋅
3183:ψ
3179:⋅
3170:ψ
3164:≠
3154:ψ
3150:⋅
3141:ψ
3119:⋅
3087:ψ
2884:⇔
2610:−
2575:−
2539:−
2320:↔
2311:↔
2302:↔
2259:→
2250:→
2237:↔
2221:→
2212:→
2172:∨
2163:↔
2154:∨
2118:∧
2109:↔
2100:∧
2030:⇔
2004:∧
1995:⇔
1986:∧
1954:∨
1945:⇔
1936:∨
1841:Tautology
1566:Egyptians
1394:≠
1152:∘
1042:∘
936:from the
809:logarithm
778:≠
736:−
727:−
718:−
688:−
682:≠
676:−
644:÷
638:≠
632:÷
501:→
486:→
471:→
456:→
399:∗
387:∗
364:∗
307:∈
284:∗
272:∗
238:∗
213:symmetric
127:∈
115:∀
108:∗
96:∗
63:Statement
4163:Symmetry
4119:8 August
4109:MacTutor
4064:8 August
4059:MacTutor
3898:Articles
3393:See also
3256:momentum
3186:′
3157:′
2931:such as
2696:symmetry
2682:Symmetry
2665:matrices
2364:addition
2356:analysis
1615:commuter
1583:Elements
1574:products
1532:; i.e.,
707:, since
619:Division
540:integers
524:Addition
421:Examples
345:or that
337:commutes
217:equality
204:addition
180:division
164:operands
76:operands
47:Property
3980:, 2001
3638:Servois
3565:Commute
3434:physics
3359:is the
3258:in the
2971:), and
2053:with".
2042:" is a
2022:where "
1914:within
1256:from a
1028:. Then
805:th-root
590:" and "
567:algebra
354:commute
57:Algebra
4031:"Yark"
3938:
3886:
3858:
3832:
3813:
3783:
3567:, p. 1
3513:
3167:
3161:
2663:or of
2423:whose
2047:symbol
1578:Euclid
1250:linear
356:under
3916:(PDF)
3909:(PDF)
3769:Books
3461:Notes
2836:then
2751:graph
2432:field
2430:In a
2419:is a
2410:group
2408:is a
2404:, or
2348:group
2051:proof
1972:and
1905:valid
1899:, or
1540:= −(
859:B ⇒ A
854:A ⇒ B
819:Some
573:Union
556:field
339:with
250:on a
53:Field
4121:2007
4089:2008
4066:2007
3936:ISBN
3884:ISBN
3856:ISBN
3830:ISBN
3811:ISBN
3781:ISBN
3589:2024
3511:ISBN
3298:and
3041:and
2590:but
2521:and
2421:ring
2366:and
2358:and
2350:and
1252:and
1138:and
984:and
835:are
831:and
581:sets
575:and
550:and
526:and
349:and
202:and
182:and
154:, a
43:Type
4038:at
3988:at
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2753:in
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1918:in
1548:).
1271:of
932:of
588:And
376:if
261:if
252:set
192:not
174:or
158:is
150:In
70:is
4144::
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3653:.
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3487:^
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3469:^
3108::
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592:or
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