47:
1526:
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2698:
437:
450:
1289:
3249:
1521:{\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}}}
2465:
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
1597:. 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.
3122:
2293:
2347:
528:
217:
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
334:
3348:
1257:
1147:
2145:
2031:
2199:
1981:
154:
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668:
2664:
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2051:
809:
3085:
3050:
3012:
712:
2845:
763:
2754:
1037:
993:
3450:
3398:
3117:
426:
3368:
3244:{\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)}
2792:
3307:
3287:
2980:
2960:
2865:
2469:
Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function
2208:
385:
259:
2365:, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as
4158:
1836:
4115:
3400:, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.
2302:
275:
3898:
336:
In other words, an operation is commutative if every two elements commute. An operation that does not satisfy the above property is called
1152:
1042:
1986:
1936:
46:
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3950:
3870:
3844:
3825:
3795:
3525:
2931:
456:
3916:
2472:
3312:
2925:
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3425:
2100:
1850:
4183:
4178:
3507:
2154:
210:
31:
4168:
4163:
3983:
189:, the property can also be used in more advanced settings. The name is needed because there are operations, such as
99:
3961:
3455:
2466:
change. In contrast, the commutative property states that the order of the terms does not affect the final result.
1887:
1691:
1637:
1605:
3265:, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear
3262:
3261:, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually
1891:
1697:
587:
3409:
1878:
1717:
1704:
838:
for the functions are different when one changes the order of the operands. For example, the truth tables for
390:
3517:
2881:
2765:
2706:
1882:
1723:
1710:
2087:
demonstrate that commutativity is a property of particular connectives. The following are truth-functional
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190:
57:
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635:
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1812:
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2036:
3015:
2935:
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1798:
1743:
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940:
774:
577:
428:
That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative.
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2442:
2420:
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2080:
1926:
1918:
1843:
1826:
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566:
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2871:
2757:
2670:. Furthermore, associativity does not imply commutativity either – for example multiplication of
2431:
2406:
2076:
1756:
679:
583:
223:
178:
3588:
2803:
721:
3805:
Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
2715:
998:
954:
624:
4173:
4015:
3946:
3894:
3888:
3866:
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1869:
1862:
1674:
1665:
1651:
1540:
715:
606:
591:
440:
The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.
262:
177:
does not change the result. It is a fundamental property of many binary operations, and many
3511:
3353:
2939:
2427:
2402:
1915:
239:
166:
78:
811:. This property leads to two different "inverse" operations of exponentiation (namely, the
3371:
2710:
2288:{\displaystyle {\big (}P\to (Q\to R){\big )}\leftrightarrow {\big (}Q\to (P\to R){\big )}}
2057:
1819:
1283:
944:
554:
219:
2771:
1636:'s article entitled "On the real nature of symbolical algebra" published in 1840 in the
1567:
3545:
3292:
3272:
3014:. These two operators do not commute as may be seen by considering the effect of their
2965:
2945:
2850:
2709:
can be directly linked to commutativity. When a commutative operation is written as a
2435:
2378:
2370:
1633:
1580:
831:
768:
562:
546:
538:
370:
244:
3661:
2297:
Commutativity of equivalence (also called the complete commutative law of equivalence)
1575:
Records of the implicit use of the commutative property go back to ancient times. The
4152:
3923:
3088:
2412:
2395:
1762:
1536:
542:
3451:
Proof that Peano's axioms imply the commutativity of the addition of natural numbers
2697:
436:
230:
is symmetric as two equal mathematical objects are equal regardless of their order.
3460:
2358:
2061:
1930:
1680:
1268:
948:
573:
4041:
3814:
1612:
when describing functions that have what is now called the commutative property.
3854:
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.
3414:
1591:
is known to have assumed the commutative property of multiplication in his book
835:
673:
598:
558:
194:
162:
4050:
4046:
4000:
3430:
2671:
2381:
on real and complex numbers) is often used (or implicitly assumed) in proofs.
2362:
1260:
602:
449:
226:
if the relation applies regardless of the order of its operands; for example,
4018:
4023:
3550:
2536:
does not affect the result), but it is not associative (since, for example,
2054:
819:
1750:
17:
3266:
2374:
1571:
The first known use of the term was in a French
Journal published in 1814
812:
534:
214:
3917:"The Mathematical Legacy of Ancient Egypt – A Response To Robert Palter"
2342:{\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}
3444:
1744:
550:
174:
86:
67:
3880:
Abstract algebra theory. Uses commutativity property throughout book.
951:
to the real numbers is almost always noncommutative. For example, let
181:
depend on it. Perhaps most familiar as a property of arithmetic, e.g.
3996:
3934:
Article describing the mathematical ability of ancient civilizations.
1724:
1588:
4092:
822:
operation), whereas multiplication only has one inverse operation.
4007:
Definition of commutativity and examples of commutative operations
2696:
1576:
1566:
448:
435:
2203:
Commutativity of implication (also called the law of permutation)
2874:
is analogous to a commutative operation, in that if a relation
523:{\displaystyle {\vec {a}}+{\vec {b}}={\vec {b}}+{\vec {a}}.}
2438:
is commutative. (Addition in a ring is always commutative.)
1813:
1771:
1737:
1692:
3289:-direction of a particle are represented by the operators
3087:(also called products of operators) on a one-dimensional
1763:
3943:
1731:
329:{\displaystyle x*y=y*x\qquad {\mbox{for all }}x,y\in S.}
3343:{\displaystyle -i\hbar {\frac {\partial }{\partial x}}}
1844:
1799:
1705:
1681:
4139:
Biography of
Francois Servois, who first used the term
3058:
3023:
2988:
1487:
1448:
1412:
1373:
1337:
1298:
302:
4107:
Page covering the earliest uses of mathematical terms
3380:
3356:
3315:
3295:
3275:
3125:
3096:
2968:
2948:
2884:
2853:
2806:
2774:
2718:
2607:
2542:
2475:
2305:
2211:
2157:
2103:
2039:
1989:
1939:
1628:, meaning "to exchange" or "to switch", a cognate of
1292:
1271:
to itself (see below for the Matrix representation).
1252:{\displaystyle (g\circ f)(x)=g(f(x))=3(2x+1)+7=6x+10}
1155:
1142:{\displaystyle (f\circ g)(x)=f(g(x))=2(3x+7)+1=6x+15}
1045:
1001:
957:
777:
724:
682:
638:
459:
393:
373:
278:
247:
102:
3374:). This is the same example except for the constant
2373:
the commutativity of well-known operations (such as
2140:{\displaystyle (P\land Q)\leftrightarrow (Q\land P)}
2026:{\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}
2701:
Graph showing the symmetry of the addition function
93:
73:
63:
53:
3863:Algebra: Abstract and Concrete, Stressing Symmetry
3813:
3392:
3362:
3342:
3301:
3281:
3243:
3111:
3079:
3044:
3006:
2974:
2954:
2908:
2859:
2839:
2786:
2748:
2678:is always associative but not always commutative.
2658:
2593:
2520:
2341:
2287:
2194:{\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}
2193:
2139:
2045:
2025:
1976:{\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}
1975:
1520:
1251:
1141:
1031:
987:
803:
757:
706:
662:
522:
420:
379:
328:
253:
148:
30:"Commutative" redirects here. For other uses, see
3724:
3722:
2445:both addition and multiplication are commutative.
1757:
1837:
1632:. The term then appeared in English in 1838. in
453:The addition of vectors is commutative, because
4057:Examples proving some noncommutative operations
1806:
1698:
209:. The idea that simple operations, such as the
4084:Article giving the history of the real numbers
3666:Transactions of the Royal Society of Edinburgh
1638:Transactions of the Royal Society of Edinburgh
1286:is almost always noncommutative, for example:
714:. However it is classified more precisely as
149:{\displaystyle x*y=y*x\quad \forall x,y\in S.}
3941:Gay, Robins R.; Shute, Charles C. D. (1987).
2280:
2252:
2242:
2214:
1851:
1820:
1718:
1711:
1616:is the feminine form of the French adjective
8:
3702:
2920:Non-commuting operators in quantum mechanics
2528:which is clearly commutative (interchanging
39:
4093:"Earliest Known Uses of Mathematical Terms"
3662:"On the real nature of symbolical algebra"
3487:
3483:
3481:
1647:
45:
38:
3887:Hurley, Patrick J.; Watson, Lori (2016).
3691:
3501:
3499:
3379:
3355:
3325:
3314:
3294:
3274:
3211:
3205:
3203:
3140:
3134:
3132:
3124:
3095:
3059:
3057:
3027:
3022:
2989:
2987:
2967:
2947:
2883:
2852:
2805:
2773:
2717:
2606:
2541:
2497:
2474:
2385:Mathematical structures and commutativity
2304:
2279:
2278:
2251:
2250:
2241:
2240:
2213:
2212:
2210:
2156:
2102:
2038:
1988:
1938:
1906:In truth-functional propositional logic,
1482:
1443:
1407:
1368:
1332:
1293:
1291:
1154:
1044:
1000:
956:
795:
782:
776:
723:
681:
637:
619:Division, subtraction, and exponentiation
506:
505:
491:
490:
476:
475:
461:
460:
458:
392:
372:
301:
277:
246:
101:
3922:(Unpublished manuscript). Archived from
3624:
2938:, physical variables are represented by
2521:{\displaystyle f(x,y)={\frac {x+y}{2}},}
1805:
1620:, which is derived from the French noun
1539:) of two vectors in three dimensions is
936:Function composition of linear functions
850:
27:Property of some mathematical operations
3764:
3752:
3740:
3728:
3612:
3477:
3387:
3357:
3322:
1868:
1861:
1787:
1664:
1657:
1650:
205:commutative, and so are referred to as
4047:Examples of non-commutative operations
3960:Translation and interpretation of the
2909:{\displaystyle aRb\Leftrightarrow bRa}
2666:). More such examples may be found in
3812:Copi, Irving M.; Cohen, Carl (2005).
3713:
2423:whose group operation is commutative.
2401:If the operation additionally has an
1259:This also applies more generally for
7:
197:, that do not have it (for example,
3893:(12th ed.). Cengage Learning.
2060:representing "can be replaced in a
1921:. The rules allow one to transpose
1600:The first recorded use of the term
663:{\displaystyle 1\div 2\neq 2\div 1}
3331:
3327:
3212:
3206:
3141:
3135:
2668:commutative non-associative magmas
222:; a binary relation is said to be
125:
25:
3839:(6e ed.). Houghton Mifflin.
1579:used the commutative property of
572:Addition is commutative in every
4114:O'Conner, J.J.; Robertson, E.F.
4064:O'Conner, J.J.; Robertson, E.F.
3820:(12th ed.). Prentice Hall.
3513:Mathematics in Victorian Britain
2659:{\displaystyle f(f(-4,0),+4)=+1}
2594:{\displaystyle f(-4,f(0,+4))=-1}
2046:{\displaystyle \Leftrightarrow }
4159:Properties of binary operations
4116:"biography of François Servois"
4091:Cabillón, Julio; Miller, Jeff.
4034:Explanation of the term commute
3890:A Concise Introduction to Logic
2794:. For example, if the function
2756:then this function is called a
2394:is a set endowed with a total,
804:{\displaystyle 2^{3}\neq 3^{2}}
300:
124:
3865:(2e ed.). Prentice Hall.
3506:Flood, Raymond; Rice, Adrian;
3106:
3100:
2926:Canonical commutation relation
2894:
2822:
2810:
2768:is symmetric across the plane
2740:
2728:
2644:
2632:
2617:
2611:
2579:
2576:
2561:
2546:
2491:
2479:
2336:
2330:
2324:
2321:
2318:
2312:
2306:
2275:
2269:
2263:
2260:
2247:
2237:
2231:
2225:
2222:
2188:
2176:
2173:
2170:
2158:
2134:
2122:
2119:
2116:
2104:
2040:
2020:
2008:
2005:
2002:
1990:
1970:
1958:
1955:
1952:
1940:
1225:
1210:
1201:
1198:
1192:
1186:
1177:
1171:
1168:
1156:
1115:
1100:
1091:
1088:
1082:
1076:
1067:
1061:
1058:
1046:
1011:
1005:
967:
961:
834:are noncommutative, since the
752:
740:
590:are commutative operations on
545:, and, in particular, between
511:
496:
481:
466:
1:
3837:Contemporary Abstract Algebra
3788:Linear Algebra Done Right, 2e
3426:Commutative (neurophysiology)
3080:{\textstyle {\frac {d}{dx}}x}
3045:{\textstyle x{\frac {d}{dx}}}
1608:in 1814, which used the word
565:. This is also true in every
173:if changing the order of the
85:if changing the order of the
3492:Commutative and Distributive
3007:{\textstyle {\frac {d}{dx}}}
2149:Commutativity of disjunction
2095:Commutativity of conjunction
2068:Truth functional connectives
32:Commutative (disambiguation)
4003:., Accessed 8 August 2007.
3984:Encyclopedia of Mathematics
3861:Goodman, Frederick (2003).
707:{\displaystyle 0-1\neq 1-0}
89:does not change the result.
4200:
4053:., Accessed 8 August 2007
4030:, Accessed 8 August 2007.
3962:Rhind Mathematical Papyrus
3647:O'Conner & Robertson,
3593:Mathematics Stack Exchange
3456:Quasi-commutative property
3415:Centralizer and normalizer
2923:
2840:{\displaystyle f(x,y)=x+y}
2685:
2458:
2398:and commutative operation.
1888:Existential generalization
1693:Biconditional introduction
758:{\displaystyle 0-1=-(1-0)}
622:
29:
4066:"History of real numbers"
3635:O'Conner & Robertson
3589:"User MathematicalOrchid"
3417:(also called a commutant)
2867:is a symmetric function.
2749:{\displaystyle z=f(x,y),}
1032:{\displaystyle g(x)=3x+7}
988:{\displaystyle f(x)=2x+1}
868:
863:
858:
853:
771:is noncommutative, since
676:is noncommutative, since
632:is noncommutative, since
614:Noncommutative operations
207:noncommutative operations
44:
3835:Gallian, Joseph (2006).
3703:Hurley & Watson 2016
3410:Anticommutative property
3393:{\displaystyle -i\hbar }
3112:{\displaystyle \psi (x)}
1879:Universal generalization
1719:Disjunction introduction
1706:Conjunction introduction
1676:Implication introduction
541:are commutative in most
421:{\displaystyle x*y=y*x.}
234:Mathematical definitions
3786:Axler, Sheldon (1997).
3660:Gregory, D. F. (1840).
3518:Oxford University Press
3372:reduced Planck constant
2766:three-dimensional space
1923:propositional variables
1535:The vector product (or
201:); such operations are
3443:(for commutativity in
3394:
3364:
3363:{\displaystyle \hbar }
3350:, respectively (where
3344:
3303:
3283:
3245:
3113:
3081:
3046:
3008:
2976:
2956:
2910:
2861:
2841:
2788:
2750:
2702:
2660:
2595:
2522:
2343:
2289:
2195:
2141:
2075:is a property of some
2047:
2027:
1977:
1738:hypothetical syllogism
1659:Propositional calculus
1583:to simplify computing
1572:
1522:
1265:affine transformations
1253:
1143:
1033:
989:
805:
759:
708:
664:
530:
524:
445:Commutative operations
441:
422:
381:
330:
255:
150:
3816:Introduction to Logic
3692:Copi & Cohen 2005
3488:Cabillón & Miller
3466:Commuting probability
3395:
3365:
3345:
3304:
3284:
3255:uncertainty principle
3246:
3114:
3082:
3047:
3009:
2977:
2962:(meaning multiply by
2957:
2911:
2862:
2842:
2789:
2751:
2700:
2688:Distributive property
2661:
2596:
2523:
2392:commutative semigroup
2344:
2290:
2196:
2142:
2048:
2028:
1978:
1780:Negation introduction
1773:modus ponendo tollens
1570:
1563:History and etymology
1523:
1280:Matrix multiplication
1275:Matrix multiplication
1254:
1144:
1034:
990:
806:
760:
709:
665:
525:
452:
439:
423:
382:
331:
256:
151:
3915:Lumpkin, B. (1997).
3625:Gay & Shute 1987
3546:"Symmetric Relation"
3378:
3354:
3313:
3293:
3273:
3123:
3094:
3056:
3021:
2986:
2966:
2946:
2882:
2851:
2804:
2772:
2716:
2605:
2540:
2473:
2461:Associative property
2303:
2209:
2155:
2101:
2085:logical equivalences
2079:of truth functional
2037:
1987:
1937:
1919:rules of replacement
1838:Material implication
1789:Rules of replacement
1652:Transformation rules
1624:and the French verb
1290:
1153:
1043:
999:
955:
941:Function composition
775:
722:
680:
636:
457:
391:
371:
276:
245:
100:
40:Commutative property
4184:Functional analysis
4179:Concepts in physics
4126:on 2 September 2009
3441:Particle statistics
3421:Commutative diagram
2878:is symmetric, then
2787:{\displaystyle y=x}
2081:propositional logic
2077:logical connectives
1927:logical expressions
1902:Rule of replacement
1751:destructive dilemma
1644:Propositional logic
1604:was in a memoir by
179:mathematical proofs
41:
4169:Rules of inference
4164:Elementary algebra
4016:Weisstein, Eric W.
3945:. British Museum.
3543:Weisstein, Eric W.
3390:
3360:
3340:
3299:
3279:
3241:
3109:
3077:
3042:
3004:
2972:
2952:
2906:
2872:symmetric relation
2857:
2837:
2784:
2758:symmetric function
2746:
2703:
2656:
2591:
2518:
2450:Related properties
2407:commutative monoid
2339:
2285:
2191:
2137:
2043:
2023:
1973:
1870:Rules of inference
1666:Rules of inference
1573:
1518:
1512:
1473:
1437:
1398:
1362:
1323:
1249:
1139:
1029:
985:
844:(B ⇒ A) = (A ∨ ¬B)
840:(A ⇒ B) = (¬A ∨ B)
818:operation and the
801:
755:
704:
660:
607:logical operations
605:" are commutative
531:
520:
442:
418:
377:
326:
306:
251:
146:
94:Symbolic statement
3900:978-1-337-51478-1
3743:, pp. 26, 87
3436:Parallelogram law
3338:
3302:{\displaystyle x}
3282:{\displaystyle x}
3253:According to the
3220:
3179:
3173:
3149:
3072:
3040:
3002:
2975:{\displaystyle x}
2955:{\displaystyle x}
2934:as formulated by
2932:quantum mechanics
2870:For relations, a
2860:{\displaystyle f}
2513:
2417:commutative group
1933:. The rules are:
1899:
1898:
931:
930:
514:
499:
484:
469:
380:{\displaystyle *}
305:
254:{\displaystyle *}
159:
158:
16:(Redirected from
4191:
4135:
4133:
4131:
4122:. Archived from
4103:
4101:
4099:
4080:
4078:
4076:
4045:
4029:
4028:
3992:
3972:Online resources
3956:
3930:
3929:on 13 July 2007.
3928:
3921:
3904:
3876:
3850:
3831:
3819:
3801:
3768:
3762:
3756:
3750:
3744:
3738:
3732:
3726:
3717:
3711:
3705:
3700:
3694:
3689:
3683:
3682:Moore and Parker
3680:
3674:
3673:
3657:
3651:
3645:
3639:
3633:
3627:
3622:
3616:
3610:
3604:
3603:
3601:
3599:
3585:
3579:
3572:
3566:
3563:
3557:
3556:
3555:
3538:
3532:
3531:
3503:
3494:
3485:
3399:
3397:
3396:
3391:
3369:
3367:
3366:
3361:
3349:
3347:
3346:
3341:
3339:
3337:
3326:
3308:
3306:
3305:
3300:
3288:
3286:
3285:
3280:
3250:
3248:
3247:
3242:
3240:
3236:
3221:
3219:
3215:
3209:
3204:
3199:
3177:
3171:
3170:
3150:
3148:
3144:
3138:
3133:
3118:
3116:
3115:
3110:
3086:
3084:
3083:
3078:
3073:
3071:
3060:
3051:
3049:
3048:
3043:
3041:
3039:
3028:
3013:
3011:
3010:
3005:
3003:
3001:
2990:
2981:
2979:
2978:
2973:
2961:
2959:
2958:
2953:
2940:linear operators
2915:
2913:
2912:
2907:
2866:
2864:
2863:
2858:
2846:
2844:
2843:
2838:
2799:
2793:
2791:
2790:
2785:
2755:
2753:
2752:
2747:
2665:
2663:
2662:
2657:
2600:
2598:
2597:
2592:
2527:
2525:
2524:
2519:
2514:
2509:
2498:
2428:commutative ring
2403:identity element
2348:
2346:
2345:
2340:
2294:
2292:
2291:
2286:
2284:
2283:
2256:
2255:
2246:
2245:
2218:
2217:
2200:
2198:
2197:
2192:
2146:
2144:
2143:
2138:
2083:. The following
2052:
2050:
2049:
2044:
2032:
2030:
2029:
2024:
1982:
1980:
1979:
1974:
1853:
1846:
1839:
1827:De Morgan's laws
1822:
1815:
1808:
1801:
1775:
1767:
1759:
1752:
1746:
1739:
1733:
1726:
1720:
1713:
1707:
1700:
1694:
1687:
1677:
1648:
1606:François Servois
1541:anti-commutative
1527:
1525:
1524:
1519:
1517:
1516:
1478:
1477:
1442:
1441:
1403:
1402:
1367:
1366:
1328:
1327:
1258:
1256:
1255:
1250:
1148:
1146:
1145:
1140:
1038:
1036:
1035:
1030:
994:
992:
991:
986:
945:linear functions
871:
866:
861:
856:
851:
845:
841:
810:
808:
807:
802:
800:
799:
787:
786:
764:
762:
761:
756:
716:anti-commutative
713:
711:
710:
705:
669:
667:
666:
661:
625:Equation xy = yx
555:rational numbers
529:
527:
526:
521:
516:
515:
507:
501:
500:
492:
486:
485:
477:
471:
470:
462:
427:
425:
424:
419:
386:
384:
383:
378:
363:
359:
355:
346:
335:
333:
332:
327:
307:
303:
260:
258:
257:
252:
240:binary operation
220:binary relations
200:
188:
184:
167:binary operation
155:
153:
152:
147:
79:binary operation
49:
42:
21:
4199:
4198:
4194:
4193:
4192:
4190:
4189:
4188:
4149:
4148:
4146:
4129:
4127:
4113:
4097:
4095:
4090:
4074:
4072:
4063:
4040:
4014:
4013:
3995:Krowne, Aaron,
3979:"Commutativity"
3977:
3974:
3953:
3940:
3926:
3919:
3914:
3911:
3901:
3886:
3873:
3860:
3847:
3834:
3828:
3811:
3798:
3785:
3782:
3777:
3772:
3771:
3763:
3759:
3751:
3747:
3739:
3735:
3727:
3720:
3712:
3708:
3701:
3697:
3690:
3686:
3681:
3677:
3659:
3658:
3654:
3646:
3642:
3634:
3630:
3623:
3619:
3611:
3607:
3597:
3595:
3587:
3586:
3582:
3573:
3569:
3564:
3560:
3541:
3540:
3539:
3535:
3528:
3510:, eds. (2011).
3505:
3504:
3497:
3486:
3479:
3474:
3406:
3376:
3375:
3352:
3351:
3330:
3311:
3310:
3291:
3290:
3271:
3270:
3226:
3222:
3210:
3192:
3163:
3139:
3121:
3120:
3092:
3091:
3064:
3054:
3053:
3032:
3019:
3018:
2994:
2984:
2983:
2964:
2963:
2944:
2943:
2928:
2922:
2880:
2879:
2849:
2848:
2802:
2801:
2795:
2770:
2769:
2714:
2713:
2711:binary function
2695:
2690:
2684:
2603:
2602:
2538:
2537:
2499:
2471:
2470:
2463:
2457:
2452:
2387:
2355:
2301:
2300:
2207:
2206:
2153:
2152:
2099:
2098:
2070:
2035:
2034:
1985:
1984:
1935:
1934:
1904:
1863:Predicate logic
1857:
1821:Double negation
1675:
1646:
1565:
1533:
1511:
1510:
1505:
1499:
1498:
1493:
1483:
1472:
1471:
1466:
1460:
1459:
1454:
1444:
1436:
1435:
1430:
1424:
1423:
1418:
1408:
1397:
1396:
1391:
1385:
1384:
1379:
1369:
1361:
1360:
1355:
1349:
1348:
1343:
1333:
1322:
1321:
1316:
1310:
1309:
1304:
1294:
1288:
1287:
1284:square matrices
1277:
1151:
1150:
1041:
1040:
997:
996:
953:
952:
938:
869:
864:
859:
854:
843:
839:
832:truth functions
828:
826:Truth functions
791:
778:
773:
772:
720:
719:
678:
677:
634:
633:
627:
621:
616:
563:complex numbers
547:natural numbers
455:
454:
447:
434:
389:
388:
369:
368:
361:
357:
351:
344:
274:
273:
243:
242:
236:
199:"3 − 5 ≠ 5 − 3"
198:
187:"2 × 5 = 5 × 2"
186:
183:"3 + 4 = 4 + 3"
182:
98:
97:
35:
28:
23:
22:
15:
12:
11:
5:
4197:
4195:
4187:
4186:
4181:
4176:
4171:
4166:
4161:
4151:
4150:
4144:
4143:
4142:
4141:
4111:
4110:
4109:
4088:
4087:
4086:
4061:
4060:
4059:
4038:
4037:
4036:
4011:
4010:
4009:
3993:
3973:
3970:
3969:
3968:
3967:
3966:
3951:
3938:
3937:
3936:
3910:
3907:
3906:
3905:
3899:
3884:
3883:
3882:
3871:
3858:
3857:
3856:
3845:
3832:
3826:
3809:
3808:
3807:
3796:
3781:
3778:
3776:
3773:
3770:
3769:
3757:
3745:
3733:
3718:
3706:
3695:
3684:
3675:
3652:
3640:
3628:
3617:
3605:
3580:
3567:
3558:
3533:
3526:
3495:
3476:
3475:
3473:
3470:
3469:
3468:
3463:
3458:
3453:
3448:
3438:
3433:
3428:
3423:
3418:
3412:
3405:
3402:
3389:
3386:
3383:
3359:
3336:
3333:
3329:
3324:
3321:
3318:
3298:
3278:
3239:
3235:
3232:
3229:
3225:
3218:
3214:
3208:
3202:
3198:
3195:
3191:
3188:
3185:
3182:
3176:
3169:
3166:
3162:
3159:
3156:
3153:
3147:
3143:
3137:
3131:
3128:
3108:
3105:
3102:
3099:
3076:
3070:
3067:
3063:
3038:
3035:
3031:
3026:
3000:
2997:
2993:
2971:
2951:
2924:Main article:
2921:
2918:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2856:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2800:is defined as
2783:
2780:
2777:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2705:Some forms of
2694:
2691:
2686:Main article:
2683:
2680:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2517:
2512:
2508:
2505:
2502:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2459:Main article:
2456:
2453:
2451:
2448:
2447:
2446:
2439:
2436:multiplication
2424:
2409:
2399:
2386:
2383:
2379:multiplication
2371:linear algebra
2354:
2351:
2350:
2349:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2298:
2295:
2282:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2254:
2249:
2244:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2216:
2204:
2201:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2150:
2147:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2096:
2069:
2066:
2042:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1945:
1942:
1931:logical proofs
1903:
1900:
1897:
1896:
1895:
1894:
1885:
1873:
1872:
1866:
1865:
1859:
1858:
1856:
1855:
1848:
1841:
1834:
1829:
1824:
1817:
1814:Distributivity
1810:
1803:
1795:
1792:
1791:
1785:
1784:
1783:
1782:
1777:
1754:
1741:
1728:
1715:
1702:
1689:
1669:
1668:
1662:
1661:
1655:
1654:
1645:
1642:
1634:Duncan Gregory
1581:multiplication
1564:
1561:
1532:
1531:Vector product
1529:
1515:
1509:
1506:
1504:
1501:
1500:
1497:
1494:
1492:
1489:
1488:
1486:
1481:
1476:
1470:
1467:
1465:
1462:
1461:
1458:
1455:
1453:
1450:
1449:
1447:
1440:
1434:
1431:
1429:
1426:
1425:
1422:
1419:
1417:
1414:
1413:
1411:
1406:
1401:
1395:
1392:
1390:
1387:
1386:
1383:
1380:
1378:
1375:
1374:
1372:
1365:
1359:
1356:
1354:
1351:
1350:
1347:
1344:
1342:
1339:
1338:
1336:
1331:
1326:
1320:
1317:
1315:
1312:
1311:
1308:
1305:
1303:
1300:
1299:
1297:
1276:
1273:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1212:
1209:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1051:
1048:
1028:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
984:
981:
978:
975:
972:
969:
966:
963:
960:
937:
934:
933:
932:
929:
928:
925:
922:
919:
915:
914:
911:
908:
905:
901:
900:
897:
894:
891:
887:
886:
883:
880:
877:
873:
872:
867:
862:
857:
827:
824:
798:
794:
790:
785:
781:
769:Exponentiation
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
703:
700:
697:
694:
691:
688:
685:
659:
656:
653:
650:
647:
644:
641:
620:
617:
615:
612:
611:
610:
595:
581:
570:
543:number systems
539:multiplication
519:
513:
510:
504:
498:
495:
489:
483:
480:
474:
468:
465:
446:
443:
433:
430:
417:
414:
411:
408:
405:
402:
399:
396:
376:
343:One says that
338:noncommutative
325:
322:
319:
316:
313:
310:
299:
296:
293:
290:
287:
284:
281:
250:
235:
232:
211:multiplication
157:
156:
145:
142:
139:
136:
133:
130:
127:
123:
120:
117:
114:
111:
108:
105:
95:
91:
90:
75:
71:
70:
65:
61:
60:
55:
51:
50:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4196:
4185:
4182:
4180:
4177:
4175:
4172:
4170:
4167:
4165:
4162:
4160:
4157:
4156:
4154:
4147:
4140:
4137:
4136:
4125:
4121:
4117:
4112:
4108:
4105:
4104:
4094:
4089:
4085:
4082:
4081:
4071:
4067:
4062:
4058:
4055:
4054:
4052:
4048:
4043:
4039:
4035:
4032:
4031:
4026:
4025:
4020:
4017:
4012:
4008:
4005:
4004:
4002:
3998:
3994:
3990:
3986:
3985:
3980:
3976:
3975:
3971:
3965:
3963:
3958:
3957:
3954:
3952:0-7141-0944-4
3948:
3944:
3939:
3935:
3932:
3931:
3925:
3918:
3913:
3912:
3908:
3902:
3896:
3892:
3891:
3885:
3881:
3878:
3877:
3874:
3872:0-13-067342-0
3868:
3864:
3859:
3855:
3852:
3851:
3848:
3846:0-618-51471-6
3842:
3838:
3833:
3829:
3827:9780131898349
3823:
3818:
3817:
3810:
3806:
3803:
3802:
3799:
3797:0-387-98258-2
3793:
3789:
3784:
3783:
3779:
3774:
3767:, p. 250
3766:
3761:
3758:
3755:, p. 236
3754:
3749:
3746:
3742:
3737:
3734:
3730:
3725:
3723:
3719:
3715:
3710:
3707:
3704:
3699:
3696:
3693:
3688:
3685:
3679:
3676:
3671:
3667:
3663:
3656:
3653:
3650:
3644:
3641:
3638:
3632:
3629:
3626:
3621:
3618:
3614:
3609:
3606:
3594:
3590:
3584:
3581:
3577:
3571:
3568:
3562:
3559:
3553:
3552:
3547:
3544:
3537:
3534:
3529:
3527:9780191627941
3523:
3520:. p. 4.
3519:
3515:
3514:
3509:
3508:Wilson, Robin
3502:
3500:
3496:
3493:
3489:
3484:
3482:
3478:
3471:
3467:
3464:
3462:
3459:
3457:
3454:
3452:
3449:
3446:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3416:
3413:
3411:
3408:
3407:
3403:
3401:
3384:
3381:
3373:
3334:
3319:
3316:
3296:
3276:
3268:
3264:
3263:complementary
3260:
3256:
3251:
3237:
3233:
3230:
3227:
3223:
3216:
3200:
3196:
3193:
3189:
3186:
3183:
3180:
3174:
3167:
3164:
3160:
3157:
3154:
3151:
3145:
3129:
3126:
3103:
3097:
3090:
3089:wave function
3074:
3068:
3065:
3061:
3036:
3033:
3029:
3024:
3017:
2998:
2995:
2991:
2969:
2949:
2941:
2937:
2933:
2927:
2919:
2917:
2903:
2900:
2897:
2891:
2888:
2885:
2877:
2873:
2868:
2854:
2834:
2831:
2828:
2825:
2819:
2816:
2813:
2807:
2798:
2781:
2778:
2775:
2767:
2763:
2759:
2743:
2737:
2734:
2731:
2725:
2722:
2719:
2712:
2708:
2699:
2692:
2689:
2681:
2679:
2677:
2673:
2669:
2653:
2650:
2647:
2641:
2638:
2635:
2629:
2626:
2623:
2620:
2614:
2608:
2588:
2585:
2582:
2573:
2570:
2567:
2564:
2558:
2555:
2552:
2549:
2543:
2535:
2531:
2515:
2510:
2506:
2503:
2500:
2494:
2488:
2485:
2482:
2476:
2467:
2462:
2455:Associativity
2454:
2449:
2444:
2440:
2437:
2433:
2429:
2425:
2422:
2418:
2414:
2413:abelian group
2410:
2408:
2404:
2400:
2397:
2393:
2389:
2388:
2384:
2382:
2380:
2376:
2372:
2368:
2364:
2360:
2352:
2333:
2327:
2315:
2309:
2299:
2296:
2272:
2266:
2257:
2234:
2228:
2219:
2205:
2202:
2185:
2182:
2179:
2167:
2164:
2161:
2151:
2148:
2131:
2128:
2125:
2113:
2110:
2107:
2097:
2094:
2093:
2092:
2090:
2086:
2082:
2078:
2074:
2073:Commutativity
2067:
2065:
2063:
2059:
2056:
2017:
2014:
2011:
1999:
1996:
1993:
1967:
1964:
1961:
1949:
1946:
1943:
1932:
1928:
1924:
1920:
1917:
1914:refer to two
1913:
1912:commutativity
1909:
1901:
1893:
1892:instantiation
1889:
1886:
1884:
1883:instantiation
1880:
1877:
1876:
1875:
1874:
1871:
1867:
1864:
1860:
1854:
1849:
1847:
1842:
1840:
1835:
1833:
1832:Transposition
1830:
1828:
1825:
1823:
1818:
1816:
1811:
1809:
1807:Commutativity
1804:
1802:
1800:Associativity
1797:
1796:
1794:
1793:
1790:
1786:
1781:
1778:
1776:
1774:
1768:
1766:
1765:modus tollens
1760:
1755:
1753:
1747:
1742:
1740:
1734:
1729:
1727:
1721:
1716:
1714:
1708:
1703:
1701:
1695:
1690:
1688:
1685:
1682:elimination (
1678:
1673:
1672:
1671:
1670:
1667:
1663:
1660:
1656:
1653:
1649:
1643:
1641:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1598:
1596:
1595:
1590:
1586:
1582:
1578:
1569:
1562:
1560:
1558:
1554:
1550:
1546:
1542:
1538:
1537:cross product
1530:
1528:
1513:
1507:
1502:
1495:
1490:
1484:
1479:
1474:
1468:
1463:
1456:
1451:
1445:
1438:
1432:
1427:
1420:
1415:
1409:
1404:
1399:
1393:
1388:
1381:
1376:
1370:
1363:
1357:
1352:
1345:
1340:
1334:
1329:
1324:
1318:
1313:
1306:
1301:
1295:
1285:
1281:
1274:
1272:
1270:
1266:
1262:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1222:
1219:
1216:
1213:
1207:
1204:
1195:
1189:
1183:
1180:
1174:
1165:
1162:
1159:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1112:
1109:
1106:
1103:
1097:
1094:
1085:
1079:
1073:
1070:
1064:
1055:
1052:
1049:
1026:
1023:
1020:
1017:
1014:
1008:
1002:
982:
979:
976:
973:
970:
964:
958:
950:
946:
942:
935:
926:
923:
920:
917:
916:
912:
909:
906:
903:
902:
898:
895:
892:
889:
888:
884:
881:
878:
875:
874:
852:
849:
848:
847:
837:
833:
825:
823:
821:
817:
815:
796:
792:
788:
783:
779:
770:
766:
749:
746:
743:
737:
734:
731:
728:
725:
717:
701:
698:
695:
692:
689:
686:
683:
675:
671:
657:
654:
651:
648:
645:
642:
639:
631:
626:
618:
613:
608:
604:
600:
596:
593:
589:
585:
582:
579:
576:and in every
575:
571:
568:
564:
560:
556:
552:
548:
544:
540:
536:
533:
532:
517:
508:
502:
493:
487:
478:
472:
463:
451:
444:
438:
431:
429:
415:
412:
409:
406:
403:
400:
397:
394:
374:
366:
354:
349:
341:
339:
323:
320:
317:
314:
311:
308:
304:for all
297:
294:
291:
288:
285:
282:
279:
271:
267:
264:
248:
241:
233:
231:
229:
225:
221:
216:
212:
208:
204:
196:
192:
180:
176:
172:
168:
164:
143:
140:
137:
134:
131:
128:
121:
118:
115:
112:
109:
106:
103:
96:
92:
88:
84:
80:
76:
72:
69:
66:
62:
59:
56:
52:
48:
43:
37:
33:
19:
4145:
4138:
4128:. Retrieved
4124:the original
4119:
4106:
4096:. Retrieved
4083:
4073:. Retrieved
4069:
4056:
4033:
4022:
4006:
3982:
3959:
3942:
3933:
3924:the original
3889:
3879:
3862:
3853:
3836:
3815:
3804:
3790:. Springer.
3787:
3765:Gallian 2006
3760:
3753:Gallian 2006
3748:
3741:Gallian 2006
3736:
3731:, p. 34
3729:Gallian 2006
3709:
3698:
3687:
3678:
3669:
3665:
3655:
3648:
3643:
3637:Real Numbers
3636:
3631:
3620:
3615:, p. 11
3613:Lumpkin 1997
3608:
3596:. Retrieved
3592:
3583:
3575:
3570:
3565:Krowne, p. 1
3561:
3549:
3536:
3512:
3491:
3461:Trace monoid
3252:
3016:compositions
2929:
2875:
2869:
2796:
2704:
2682:Distributive
2533:
2529:
2468:
2464:
2416:
2405:, we have a
2356:
2072:
2071:
1911:
1907:
1905:
1890: /
1881: /
1772:
1769: /
1764:
1761: /
1748: /
1745:Constructive
1735: /
1722: /
1709: /
1696: /
1684:modus ponens
1683:
1679: /
1629:
1625:
1621:
1617:
1613:
1610:commutatives
1609:
1601:
1599:
1593:
1574:
1556:
1552:
1548:
1544:
1534:
1278:
1269:vector space
949:real numbers
939:
836:truth tables
829:
813:
767:
672:
628:
588:intersection
574:vector space
559:real numbers
364:
352:
347:
342:
337:
269:
265:
237:
206:
202:
170:
160:
82:
36:
4098:22 November
3997:Commutative
3716:, p. 2
3574:Weisstein,
2936:Schrödinger
2672:quaternions
2396:associative
2089:tautologies
2055:metalogical
1908:commutation
1845:Exportation
1732:Disjunctive
1725:elimination
1712:elimination
1699:elimination
1622:commutation
1614:Commutative
1602:commutative
674:Subtraction
270:commutative
195:subtraction
171:commutative
163:mathematics
83:commutative
18:Commutative
4153:Categories
4051:PlanetMath
4001:PlanetMath
3775:References
3714:Axler 1997
3672:: 208–216.
3598:20 January
3431:Commutator
3259:Heisenberg
2760:, and its
2363:set theory
2353:Set theory
1758:Absorption
1630:to commute
1618:commutatif
623:See also:
268:is called
4024:MathWorld
4019:"Commute"
3989:EMS Press
3551:MathWorld
3388:ℏ
3382:−
3358:ℏ
3332:∂
3328:∂
3323:ℏ
3317:−
3234:ψ
3231:⋅
3194:ψ
3190:⋅
3181:ψ
3175:≠
3165:ψ
3161:⋅
3152:ψ
3130:⋅
3098:ψ
2895:⇔
2621:−
2586:−
2550:−
2331:↔
2322:↔
2313:↔
2270:→
2261:→
2248:↔
2232:→
2223:→
2183:∨
2174:↔
2165:∨
2129:∧
2120:↔
2111:∧
2041:⇔
2015:∧
2006:⇔
1997:∧
1965:∨
1956:⇔
1947:∨
1852:Tautology
1577:Egyptians
1405:≠
1163:∘
1053:∘
947:from the
820:logarithm
789:≠
747:−
738:−
729:−
699:−
693:≠
687:−
655:÷
649:≠
643:÷
512:→
497:→
482:→
467:→
410:∗
398:∗
375:∗
318:∈
295:∗
283:∗
249:∗
224:symmetric
138:∈
126:∀
119:∗
107:∗
74:Statement
4174:Symmetry
4130:8 August
4120:MacTutor
4075:8 August
4070:MacTutor
3909:Articles
3404:See also
3267:momentum
3197:′
3168:′
2942:such as
2707:symmetry
2693:Symmetry
2676:matrices
2375:addition
2367:analysis
1626:commuter
1594:Elements
1585:products
1543:; i.e.,
718:, since
630:Division
551:integers
535:Addition
432:Examples
356:or that
348:commutes
228:equality
215:addition
191:division
175:operands
87:operands
58:Property
3991:, 2001
3649:Servois
3576:Commute
3445:physics
3370:is the
3269:in the
2982:), and
2064:with".
2053:" is a
2033:where "
1925:within
1267:from a
1039:. Then
816:th-root
601:" and "
578:algebra
365:commute
68:Algebra
4042:"Yark"
3949:
3897:
3869:
3843:
3824:
3794:
3578:, p. 1
3524:
3178:
3172:
2674:or of
2434:whose
2058:symbol
1589:Euclid
1261:linear
367:under
3927:(PDF)
3920:(PDF)
3780:Books
3472:Notes
2847:then
2762:graph
2443:field
2441:In a
2430:is a
2421:group
2419:is a
2415:, or
2359:group
2062:proof
1983:and
1916:valid
1910:, or
1551:= −(
870:B ⇒ A
865:A ⇒ B
830:Some
584:Union
567:field
350:with
261:on a
64:Field
4132:2007
4100:2008
4077:2007
3947:ISBN
3895:ISBN
3867:ISBN
3841:ISBN
3822:ISBN
3792:ISBN
3600:2024
3522:ISBN
3309:and
3052:and
2601:but
2532:and
2432:ring
2377:and
2369:and
2361:and
1263:and
1149:and
995:and
846:are
842:and
592:sets
586:and
561:and
537:and
360:and
213:and
193:and
165:, a
54:Type
4049:at
3999:at
3257:of
2930:In
2764:in
2411:An
2357:In
1929:in
1559:).
1282:of
943:of
599:And
387:if
272:if
263:set
203:not
185:or
169:is
161:In
81:is
4155::
4118:.
4068:.
4021:.
3987:,
3981:,
3721:^
3670:14
3668:.
3664:.
3591:.
3548:.
3516:.
3498:^
3490:,
3480:^
3119::
2916:.
2426:A
2390:A
2091:.
1640:.
1587:.
1555:×
1547:×
1247:10
1137:15
765:.
670:.
603:or
557:,
553:,
549:,
340:.
238:A
77:A
4134:.
4102:.
4079:.
4044:.
4027:.
3964:.
3955:.
3903:.
3875:.
3849:.
3830:.
3800:.
3602:.
3554:.
3530:.
3447:)
3385:i
3335:x
3320:i
3297:x
3277:x
3238:)
3228:x
3224:(
3217:x
3213:d
3207:d
3201:=
3187:x
3184:+
3158:x
3155:=
3146:x
3142:d
3136:d
3127:x
3107:)
3104:x
3101:(
3075:x
3069:x
3066:d
3062:d
3037:x
3034:d
3030:d
3025:x
2999:x
2996:d
2992:d
2970:x
2950:x
2904:a
2901:R
2898:b
2892:b
2889:R
2886:a
2876:R
2855:f
2835:y
2832:+
2829:x
2826:=
2823:)
2820:y
2817:,
2814:x
2811:(
2808:f
2797:f
2782:x
2779:=
2776:y
2744:,
2741:)
2738:y
2735:,
2732:x
2729:(
2726:f
2723:=
2720:z
2654:1
2651:+
2648:=
2645:)
2642:4
2639:+
2636:,
2633:)
2630:0
2627:,
2624:4
2618:(
2615:f
2612:(
2609:f
2589:1
2583:=
2580:)
2577:)
2574:4
2571:+
2568:,
2565:0
2562:(
2559:f
2556:,
2553:4
2547:(
2544:f
2534:y
2530:x
2516:,
2511:2
2507:y
2504:+
2501:x
2495:=
2492:)
2489:y
2486:,
2483:x
2480:(
2477:f
2337:)
2334:P
2328:Q
2325:(
2319:)
2316:Q
2310:P
2307:(
2281:)
2276:)
2273:R
2267:P
2264:(
2258:Q
2253:(
2243:)
2238:)
2235:R
2229:Q
2226:(
2220:P
2215:(
2189:)
2186:P
2180:Q
2177:(
2171:)
2168:Q
2162:P
2159:(
2135:)
2132:P
2126:Q
2123:(
2117:)
2114:Q
2108:P
2105:(
2021:)
2018:P
2012:Q
2009:(
2003:)
2000:Q
1994:P
1991:(
1971:)
1968:P
1962:Q
1959:(
1953:)
1950:Q
1944:P
1941:(
1686:)
1557:b
1553:a
1549:a
1545:b
1514:]
1508:1
1503:0
1496:1
1491:0
1485:[
1480:=
1475:]
1469:1
1464:0
1457:1
1452:1
1446:[
1439:]
1433:1
1428:0
1421:1
1416:0
1410:[
1400:]
1394:1
1389:0
1382:1
1377:0
1371:[
1364:]
1358:1
1353:0
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1335:[
1330:=
1325:]
1319:1
1314:0
1307:2
1302:0
1296:[
1244:+
1241:x
1238:6
1235:=
1232:7
1229:+
1226:)
1223:1
1220:+
1217:x
1214:2
1211:(
1208:3
1205:=
1202:)
1199:)
1196:x
1193:(
1190:f
1187:(
1184:g
1181:=
1178:)
1175:x
1172:(
1169:)
1166:f
1160:g
1157:(
1134:+
1131:x
1128:6
1125:=
1122:1
1119:+
1116:)
1113:7
1110:+
1107:x
1104:3
1101:(
1098:2
1095:=
1092:)
1089:)
1086:x
1083:(
1080:g
1077:(
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1071:=
1068:)
1065:x
1062:(
1059:)
1056:g
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1027:7
1024:+
1021:x
1018:3
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1009:x
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980:+
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968:)
965:x
962:(
959:f
927:T
924:T
921:T
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913:T
910:F
907:F
904:T
899:F
896:T
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882:T
879:F
876:F
860:B
855:A
814:n
797:2
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750:0
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735:=
732:1
726:0
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609:.
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413:x
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401:y
395:x
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289:=
286:y
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266:S
144:.
141:S
135:y
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116:y
113:=
110:y
104:x
34:.
20:)
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