1640:
1627:
1061:
1051:
1021:
1011:
989:
979:
944:
922:
912:
877:
845:
835:
810:
778:
763:
743:
711:
701:
691:
671:
639:
619:
609:
599:
567:
547:
503:
483:
473:
436:
411:
401:
369:
344:
307:
277:
245:
183:
148:
3836:
in which each odd number is related to itself and there are no other relations. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation. The union of a coreflexive relation and a transitive relation on
3824:
is "has the same limit as" on the set of sequences of real numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the same limit as some sequence, then it has the same limit as itself. An example of a left quasi-reflexive relation is a left
3645:
3753:. Not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but others are not (that is, neither all nor none are). For example, the binary relation "the product of
3654:
1371:
1300:
1615:
1229:
2502:
1500:
1456:
1412:
2095:
1637:
indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by
1570:
1467:
1423:
1382:
1311:
1240:
1184:
2172:
2237:
3190:
3275:
3603:
1173:
1559:
3906:
2590:
2375:
2257:
2023:
2800:
1306:
3347:
3232:
3050:
2949:
2848:
2766:
3747:
1985:
1713:
1235:
2653:
2630:
2610:
3405:
3376:
3304:
3134:
3079:
3007:
2978:
2906:
2877:
2565:
1794:
3629:
3535:
3509:
3105:
2737:
1765:
1739:
1527:
1144:
2545:
2418:
2304:
3863:
3822:
3791:
3771:
3559:
3479:
3455:
3425:
3154:
2680:
2522:
2395:
2355:
2324:
2277:
2201:
2139:
2119:
1955:
1935:
1876:
1852:
1829:
1674:
48:
1565:
1179:
4625:
2423:
4468:
Fonseca de
Oliveira, José Nuno; Pereira Cunha Rodrigues, César de Jesus (2004), "Transposing relations: from Maybe functions to hash tables",
1462:
4581:
1418:
1377:
4295:
3136:
Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation
4519:
4545:
4527:
4505:
4486:
4458:
3723:
An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (
2031:
3829:, which is always left quasi-reflexive but not necessarily right quasi-reflexive, and thus not necessarily quasi-reflexive.
41:
4608:
4630:
4383:
2144:
4603:
2206:
3162:
34:
3237:
3668:
3568:
3428:
3196:
3020:
if every element that is part of some relation is related to itself. Explicitly, this means that whenever
2773:
1882:
1078:
71:
1149:
3948:
1088:
81:
1533:
3963:
3953:
1906:
1654:
865:
140:
4307:
3928:
3458:
2803:
1902:
1677:
1113:
524:
106:
23:
3868:
2573:
1366:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
4598:
3938:
3826:
1898:
1083:
1073:
999:
76:
66:
2360:
2242:
1990:
2779:
1295:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
4577:
4541:
4523:
4501:
4482:
4454:
3320:
3205:
3157:
3023:
2922:
2821:
2524:
can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of
2179:
2098:
1832:
3797:, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of
2742:
1633:
indicates that the column's property is always true the row's term (at the very left), while
4554:
3726:
2806:
is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
1964:
1683:
798:
731:
2635:
2615:
2595:
4310:
often use different terminology. Reflexive relations in the mathematical sense are called
3923:
3381:
3352:
3280:
3110:
3055:
2983:
2954:
2882:
2853:
2550:
1809:
1770:
659:
456:
27:
3608:
3514:
3488:
3084:
2716:
1797:
A term's definition may require additional properties that are not listed in this table.
1744:
1718:
1506:
1123:
2527:
2400:
2286:
1889:, since every real number is equal to itself. A reflexive relation is said to have the
3848:
3807:
3798:
3776:
3756:
3544:
3538:
3464:
3440:
3410:
3139:
2769:
2665:
2507:
2380:
2340:
2309:
2262:
2186:
2124:
2104:
1940:
1920:
1861:
1837:
1814:
1659:
327:
4619:
3710:
1098:
1093:
932:
265:
91:
86:
3644:
4558:
3958:
3794:
3750:
2568:
1886:
1805:
389:
2662:
There are several definitions related to the reflexive property. The relation
1610:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
4358:
1224:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
587:
4451:
Deductive Logic – An
Introduction to Evaluation Techniques and Logical Theory
2497:{\displaystyle R\setminus \operatorname {I} _{X}=\{(x,y)\in R~:~x\neq y\}.}
4395:
3943:
2280:
204:
3653:
1495:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
3833:
3703:
3682:
2547:
For example, the reflexive closure of the canonical strict inequality
1451:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
3675:
1407:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
2326:
is reflexive if and only if it is equal to its reflexive closure.
2239:
which can equivalently be defined as the smallest (with respect to
4314:
in philosophical logic, and quasi-reflexive relations are called
4226:
4540:, Revised Edition, Reprinted 2003, Harvard University Press,
4408:
4365:, p. 32. Russell also introduces two equivalent terms
4271:
4266:
4261:
4256:
4251:
4246:
4241:
4236:
4231:
2090:{\displaystyle \operatorname {I} _{X}:=\{(x,x)~:~x\in X\}}
2713:
if it does not relate any element to itself; that is, if
3832:
An example of a coreflexive relation is the relation on
3427:
is coreflexive if and only if its symmetric closure is
4566:(2nd ed.). London: George Allen & Unwin, Ltd.
4396:
Fonseca de
Oliveira & Pereira Cunha Rodrigues 2004
3871:
3851:
3810:
3779:
3759:
3729:
3611:
3571:
3547:
3517:
3491:
3467:
3443:
3413:
3384:
3355:
3323:
3283:
3240:
3208:
3165:
3142:
3113:
3087:
3058:
3026:
2986:
2957:
2925:
2885:
2856:
2824:
2782:
2745:
2719:
2668:
2638:
2618:
2598:
2576:
2553:
2530:
2510:
2426:
2403:
2383:
2363:
2343:
2312:
2289:
2265:
2245:
2209:
2189:
2147:
2127:
2107:
2034:
1993:
1967:
1943:
1923:
1864:
1840:
1817:
1773:
1747:
1721:
1686:
1662:
1568:
1536:
1509:
1465:
1421:
1380:
1309:
1238:
1182:
1152:
1126:
4477:
Hausman, Alan; Kahane, Howard; Tidman, Paul (2013).
1881:
An example of a reflexive relation is the relation "
16:
Binary relation that relates every element to itself
4419:
3900:
3857:
3816:
3785:
3765:
3741:
3623:
3597:
3553:
3529:
3503:
3473:
3449:
3419:
3399:
3370:
3341:
3298:
3269:
3226:
3184:
3148:
3128:
3099:
3073:
3044:
3001:
2972:
2943:
2900:
2871:
2842:
2794:
2760:
2731:
2674:
2647:
2624:
2604:
2584:
2559:
2539:
2516:
2496:
2412:
2389:
2369:
2349:
2318:
2298:
2271:
2251:
2231:
2195:
2166:
2133:
2113:
2089:
2017:
1979:
1949:
1929:
1905:, reflexivity is one of three properties defining
1870:
1846:
1823:
1788:
1759:
1733:
1707:
1668:
1609:
1553:
1521:
1494:
1450:
1406:
1365:
1294:
1223:
1167:
1138:
2167:{\displaystyle \operatorname {I} _{X}\subseteq R}
1385:
4500:, Perspectives in Mathematical Logic, Dover,
4431:
3915:-element binary relations of different types
2232:{\displaystyle R\cup \operatorname {I} _{X},}
1650:in the "Antisymmetric" column, respectively.
42:
8:
4479:Logic and Philosophy – A Modern Introduction
3185:{\displaystyle R\cup R^{\operatorname {T} }}
2488:
2446:
2084:
2048:
3695:Examples of irreflexive relations include:
4407:On-Line Encyclopedia of Integer Sequences
49:
35:
18:
3881:
3876:
3870:
3850:
3809:
3804:An example of a quasi-reflexive relation
3778:
3758:
3728:
3663:Examples of reflexive relations include:
3610:
3581:
3570:
3546:
3516:
3490:
3466:
3442:
3412:
3383:
3354:
3322:
3282:
3250:
3239:
3207:
3176:
3164:
3141:
3112:
3086:
3057:
3025:
2985:
2956:
2924:
2884:
2855:
2823:
2781:
2744:
2718:
2667:
2637:
2617:
2597:
2578:
2577:
2575:
2552:
2529:
2509:
2437:
2425:
2402:
2382:
2362:
2342:
2311:
2288:
2264:
2244:
2220:
2208:
2188:
2152:
2146:
2126:
2106:
2039:
2033:
1992:
1966:
1942:
1922:
1863:
1839:
1816:
1772:
1746:
1720:
1685:
1661:
1589:
1569:
1567:
1537:
1535:
1508:
1483:
1466:
1464:
1439:
1422:
1420:
1395:
1381:
1379:
1343:
1320:
1310:
1308:
1281:
1252:
1239:
1237:
1204:
1183:
1181:
1151:
1125:
3909:
3845:The number of reflexive relations on an
4560:Introduction to Mathematical Philosophy
4449:Clarke, D.S.; Behling, Richard (1998).
4362:
4346:
4327:
3437:A reflexive relation on a nonempty set
2430:
2397:that has the same reflexive closure as
1159:
32:
4386:calls this property quasi-reflexivity.
3156:is quasi-reflexive if and only if its
3270:{\displaystyle xRy{\text{ and }}yRx,}
7:
4568:(Online corrected edition, Feb 2010)
4334:
3793:is even" is reflexive on the set of
3598:{\displaystyle xRy{\text{ and }}yRz}
1653:All definitions tacitly require the
4470:Mathematics of Program Construction
4296:Stirling numbers of the second kind
3837:the same set is always transitive.
3192:is left (or right) quasi-reflexive.
2612:whereas the reflexive reduction of
2592:is the usual non-strict inequality
1168:{\displaystyle S\neq \varnothing :}
4520:Undergraduate Texts in Mathematics
3177:
2434:
2217:
2149:
2036:
14:
4420:Hausman, Kahane & Tidman 2013
3706:to" on the integers larger than 1
2357:is the smallest (with respect to
3652:
3643:
3457:can neither be irreflexive, nor
1638:
1625:
1554:{\displaystyle {\text{not }}aRa}
1108:
1059:
1049:
1019:
1009:
987:
977:
942:
920:
910:
875:
843:
833:
808:
776:
761:
741:
709:
699:
689:
669:
637:
617:
607:
597:
565:
545:
501:
481:
471:
434:
409:
399:
367:
342:
305:
275:
243:
181:
146:
101:
4453:. University Press of America.
1858:if it relates every element of
4626:Properties of binary relations
4576:, Cambridge University Press,
2461:
2449:
2063:
2051:
2006:
1994:
1646:in the "Symmetric" column and
1582:
1327:
1272:
1201:
1:
3841:Number of reflexive relations
3688:"is greater than or equal to"
1647:
1634:
1044:
1039:
1034:
1029:
1004:
972:
967:
962:
957:
952:
937:
905:
900:
895:
890:
885:
870:
858:
853:
828:
823:
818:
803:
791:
786:
771:
756:
751:
736:
724:
719:
684:
679:
664:
652:
647:
632:
627:
592:
580:
575:
560:
555:
540:
535:
530:
516:
511:
496:
491:
466:
461:
449:
444:
429:
424:
419:
394:
382:
377:
362:
357:
352:
337:
332:
320:
315:
300:
295:
290:
285:
270:
258:
253:
238:
233:
228:
223:
218:
213:
196:
191:
176:
171:
166:
161:
156:
3901:{\displaystyle 2^{n^{2}-n}.}
2585:{\displaystyle \mathbb {R} }
4604:Encyclopedia of Mathematics
4514:Lidl, R.; Pilz, G. (1998),
2504:The reflexive reduction of
4649:
3691:"is less than or equal to"
2768:A relation is irreflexive
2370:{\displaystyle \subseteq }
2252:{\displaystyle \subseteq }
2018:{\displaystyle (x,x)\in R}
4572:Schmidt, Gunther (2010),
4432:Clarke & Behling 1998
2795:{\displaystyle X\times X}
4516:Applied abstract algebra
3342:{\displaystyle x,y\in X}
3227:{\displaystyle x,y\in X}
3045:{\displaystyle x,y\in X}
2944:{\displaystyle x,y\in X}
2843:{\displaystyle x,y\in X}
2259:) reflexive relation on
4384:Encyclopedia Britannica
2761:{\displaystyle x\in X.}
4574:Relational Mathematics
3902:
3859:
3818:
3787:
3767:
3743:
3742:{\displaystyle x>y}
3625:
3599:
3555:
3531:
3505:
3475:
3451:
3421:
3401:
3372:
3343:
3300:
3271:
3228:
3186:
3150:
3130:
3101:
3075:
3046:
3003:
2974:
2945:
2902:
2873:
2844:
2796:
2762:
2733:
2676:
2649:
2626:
2606:
2586:
2561:
2541:
2518:
2498:
2414:
2391:
2371:
2351:
2320:
2300:
2273:
2253:
2233:
2197:
2168:
2135:
2115:
2091:
2028:Equivalently, letting
2019:
1981:
1980:{\displaystyle x\in X}
1951:
1931:
1893:or is said to possess
1872:
1848:
1825:
1790:
1761:
1735:
1709:
1708:{\displaystyle a,b,c,}
1670:
1611:
1555:
1523:
1496:
1452:
1408:
1367:
1296:
1225:
1169:
1140:
4536:Quine, W. V. (1951),
3903:
3860:
3819:
3788:
3768:
3744:
3626:
3600:
3556:
3532:
3506:
3476:
3452:
3422:
3402:
3373:
3344:
3301:
3272:
3229:
3187:
3151:
3131:
3102:
3076:
3047:
3004:
2975:
2946:
2913:right quasi-reflexive
2903:
2874:
2845:
2797:
2763:
2734:
2677:
2650:
2648:{\displaystyle <.}
2627:
2625:{\displaystyle \leq }
2607:
2605:{\displaystyle \leq }
2587:
2562:
2542:
2519:
2499:
2415:
2392:
2372:
2352:
2321:
2301:
2274:
2254:
2234:
2198:
2169:
2136:
2116:
2092:
2020:
1982:
1952:
1932:
1907:equivalence relations
1873:
1849:
1826:
1791:
1762:
1736:
1710:
1671:
1612:
1556:
1524:
1497:
1453:
1409:
1368:
1297:
1226:
1170:
1141:
1120:Definitions, for all
4357:This term is due to
3964:Equivalence relation
3869:
3849:
3808:
3777:
3757:
3727:
3609:
3569:
3545:
3515:
3489:
3465:
3441:
3411:
3400:{\displaystyle x=y.}
3382:
3371:{\displaystyle xRy,}
3353:
3321:
3299:{\displaystyle x=y.}
3281:
3238:
3206:
3163:
3140:
3129:{\displaystyle yRy.}
3111:
3085:
3074:{\displaystyle xRy,}
3056:
3024:
3002:{\displaystyle yRy.}
2984:
2973:{\displaystyle xRy,}
2955:
2923:
2901:{\displaystyle xRx.}
2883:
2872:{\displaystyle xRy,}
2854:
2822:
2812:left quasi-reflexive
2780:
2743:
2717:
2666:
2636:
2616:
2596:
2574:
2560:{\displaystyle <}
2551:
2528:
2508:
2424:
2401:
2381:
2361:
2341:
2310:
2287:
2263:
2243:
2207:
2187:
2145:
2125:
2105:
2032:
1991:
1965:
1941:
1921:
1862:
1838:
1815:
1789:{\displaystyle aRc.}
1771:
1745:
1719:
1684:
1660:
1655:homogeneous relation
1566:
1534:
1507:
1463:
1419:
1378:
1307:
1236:
1180:
1150:
1124:
866:Strict partial order
141:Equivalence relation
4631:Reflexive relations
4522:, Springer-Verlag,
4472:, Springer: 334–356
4308:philosophical logic
4302:Philosophical logic
3916:
3678:of" (set inclusion)
3624:{\displaystyle xRz}
3530:{\displaystyle yRx}
3504:{\displaystyle xRy}
3100:{\displaystyle xRx}
2804:asymmetric relation
2732:{\displaystyle xRx}
2658:Related definitions
2331:reflexive reduction
1760:{\displaystyle bRc}
1734:{\displaystyle aRb}
1522:{\displaystyle aRa}
1139:{\displaystyle a,b}
525:Well-quasi-ordering
4538:Mathematical Logic
4422:, pp. 327–328
4367:to be contained in
3910:
3898:
3855:
3827:Euclidean relation
3814:
3783:
3763:
3739:
3621:
3595:
3551:
3527:
3501:
3471:
3447:
3417:
3397:
3368:
3339:
3296:
3267:
3224:
3182:
3146:
3126:
3097:
3071:
3042:
2999:
2970:
2941:
2898:
2869:
2840:
2792:
2758:
2729:
2672:
2645:
2622:
2602:
2582:
2557:
2540:{\displaystyle R.}
2537:
2514:
2494:
2413:{\displaystyle R.}
2410:
2387:
2367:
2347:
2335:irreflexive kernel
2316:
2299:{\displaystyle R.}
2296:
2269:
2249:
2229:
2193:
2164:
2131:
2111:
2087:
2015:
1977:
1947:
1927:
1891:reflexive property
1868:
1844:
1821:
1786:
1757:
1731:
1705:
1666:
1607:
1605:
1551:
1519:
1492:
1490:
1448:
1446:
1404:
1402:
1363:
1361:
1292:
1290:
1221:
1219:
1165:
1136:
1000:Strict total order
4583:978-0-521-76268-7
4555:Russell, Bertrand
4496:Levy, A. (1979),
4312:totally reflexive
4277:
4276:
3858:{\displaystyle n}
3817:{\displaystyle R}
3786:{\displaystyle y}
3766:{\displaystyle x}
3716:"is greater than"
3699:"is not equal to"
3584:
3554:{\displaystyle R}
3474:{\displaystyle R}
3450:{\displaystyle X}
3420:{\displaystyle R}
3378:then necessarily
3277:then necessarily
3253:
3158:symmetric closure
3149:{\displaystyle R}
3081:then necessarily
2980:then necessarily
2879:then necessarily
2802:is reflexive. An
2675:{\displaystyle R}
2517:{\displaystyle R}
2478:
2472:
2390:{\displaystyle X}
2350:{\displaystyle R}
2319:{\displaystyle R}
2272:{\displaystyle X}
2196:{\displaystyle R}
2180:reflexive closure
2134:{\displaystyle R}
2114:{\displaystyle X}
2099:identity relation
2074:
2068:
1950:{\displaystyle X}
1930:{\displaystyle R}
1871:{\displaystyle X}
1847:{\displaystyle X}
1824:{\displaystyle R}
1802:
1801:
1669:{\displaystyle R}
1620:
1619:
1592:
1540:
1486:
1442:
1398:
1346:
1255:
933:Strict weak order
119:Total, Semiconnex
4638:
4612:
4586:
4567:
4565:
4550:
4532:
4510:
4498:Basic Set Theory
4492:
4473:
4464:
4435:
4429:
4423:
4417:
4411:
4405:
4399:
4393:
4387:
4380:
4374:
4355:
4349:
4344:
4338:
4332:
4293:
4221:
4208:
4207:
4187:
4170:
4169:
4118:
4113:
4108:
4103:
3917:
3907:
3905:
3904:
3899:
3894:
3893:
3886:
3885:
3865:-element set is
3864:
3862:
3861:
3856:
3823:
3821:
3820:
3815:
3792:
3790:
3789:
3784:
3772:
3770:
3769:
3764:
3748:
3746:
3745:
3740:
3656:
3647:
3630:
3628:
3627:
3622:
3604:
3602:
3601:
3596:
3585:
3582:
3560:
3558:
3557:
3552:
3536:
3534:
3533:
3528:
3510:
3508:
3507:
3502:
3480:
3478:
3477:
3472:
3456:
3454:
3453:
3448:
3426:
3424:
3423:
3418:
3406:
3404:
3403:
3398:
3377:
3375:
3374:
3369:
3348:
3346:
3345:
3340:
3313:
3312:
3305:
3303:
3302:
3297:
3276:
3274:
3273:
3268:
3254:
3251:
3233:
3231:
3230:
3225:
3191:
3189:
3188:
3183:
3181:
3180:
3155:
3153:
3152:
3147:
3135:
3133:
3132:
3127:
3106:
3104:
3103:
3098:
3080:
3078:
3077:
3072:
3051:
3049:
3048:
3043:
3016:
3015:
3008:
3006:
3005:
3000:
2979:
2977:
2976:
2971:
2950:
2948:
2947:
2942:
2915:
2914:
2907:
2905:
2904:
2899:
2878:
2876:
2875:
2870:
2849:
2847:
2846:
2841:
2814:
2813:
2801:
2799:
2798:
2793:
2767:
2765:
2764:
2759:
2738:
2736:
2735:
2730:
2709:
2708:
2701:
2700:
2693:
2692:
2681:
2679:
2678:
2673:
2654:
2652:
2651:
2646:
2631:
2629:
2628:
2623:
2611:
2609:
2608:
2603:
2591:
2589:
2588:
2583:
2581:
2566:
2564:
2563:
2558:
2546:
2544:
2543:
2538:
2523:
2521:
2520:
2515:
2503:
2501:
2500:
2495:
2476:
2470:
2442:
2441:
2419:
2417:
2416:
2411:
2396:
2394:
2393:
2388:
2376:
2374:
2373:
2368:
2356:
2354:
2353:
2348:
2325:
2323:
2322:
2317:
2305:
2303:
2302:
2297:
2278:
2276:
2275:
2270:
2258:
2256:
2255:
2250:
2238:
2236:
2235:
2230:
2225:
2224:
2202:
2200:
2199:
2194:
2173:
2171:
2170:
2165:
2157:
2156:
2141:is reflexive if
2140:
2138:
2137:
2132:
2120:
2118:
2117:
2112:
2096:
2094:
2093:
2088:
2072:
2066:
2044:
2043:
2024:
2022:
2021:
2016:
1986:
1984:
1983:
1978:
1956:
1954:
1953:
1948:
1936:
1934:
1933:
1928:
1885:" on the set of
1877:
1875:
1874:
1869:
1853:
1851:
1850:
1845:
1830:
1828:
1827:
1822:
1795:
1793:
1792:
1787:
1766:
1764:
1763:
1758:
1740:
1738:
1737:
1732:
1714:
1712:
1711:
1706:
1675:
1673:
1672:
1667:
1649:
1645:
1642:
1641:
1636:
1632:
1629:
1628:
1616:
1614:
1613:
1608:
1606:
1593:
1590:
1560:
1558:
1557:
1552:
1541:
1538:
1528:
1526:
1525:
1520:
1501:
1499:
1498:
1493:
1491:
1487:
1484:
1457:
1455:
1454:
1449:
1447:
1443:
1440:
1413:
1411:
1410:
1405:
1403:
1399:
1396:
1372:
1370:
1369:
1364:
1362:
1347:
1344:
1321:
1301:
1299:
1298:
1293:
1291:
1282:
1256:
1253:
1230:
1228:
1227:
1222:
1220:
1205:
1186:
1174:
1172:
1171:
1166:
1145:
1143:
1142:
1137:
1066:
1063:
1062:
1056:
1053:
1052:
1046:
1041:
1036:
1031:
1026:
1023:
1022:
1016:
1013:
1012:
1006:
994:
991:
990:
984:
981:
980:
974:
969:
964:
959:
954:
949:
946:
945:
939:
927:
924:
923:
917:
914:
913:
907:
902:
897:
892:
887:
882:
879:
878:
872:
860:
855:
850:
847:
846:
840:
837:
836:
830:
825:
820:
815:
812:
811:
805:
799:Meet-semilattice
793:
788:
783:
780:
779:
773:
768:
765:
764:
758:
753:
748:
745:
744:
738:
732:Join-semilattice
726:
721:
716:
713:
712:
706:
703:
702:
696:
693:
692:
686:
681:
676:
673:
672:
666:
654:
649:
644:
641:
640:
634:
629:
624:
621:
620:
614:
611:
610:
604:
601:
600:
594:
582:
577:
572:
569:
568:
562:
557:
552:
549:
548:
542:
537:
532:
527:
518:
513:
508:
505:
504:
498:
493:
488:
485:
484:
478:
475:
474:
468:
463:
451:
446:
441:
438:
437:
431:
426:
421:
416:
413:
412:
406:
403:
402:
396:
384:
379:
374:
371:
370:
364:
359:
354:
349:
346:
345:
339:
334:
322:
317:
312:
309:
308:
302:
297:
292:
287:
282:
279:
278:
272:
260:
255:
250:
247:
246:
240:
235:
230:
225:
220:
215:
210:
208:
198:
193:
188:
185:
184:
178:
173:
168:
163:
158:
153:
150:
149:
143:
61:
60:
51:
44:
37:
30:
28:binary relations
19:
4648:
4647:
4641:
4640:
4639:
4637:
4636:
4635:
4616:
4615:
4597:
4594:
4589:
4584:
4571:
4563:
4553:
4548:
4535:
4530:
4513:
4508:
4495:
4489:
4476:
4467:
4461:
4448:
4444:
4439:
4438:
4430:
4426:
4418:
4414:
4406:
4402:
4394:
4390:
4381:
4377:
4371:imply diversity
4356:
4352:
4345:
4341:
4333:
4329:
4324:
4304:
4280:
4206:
4200:
4199:
4198:
4196:
4168:
4162:
4161:
4160:
4158:
4116:
4111:
4106:
4101:
3920:Elements
3877:
3872:
3867:
3866:
3847:
3846:
3843:
3806:
3805:
3799:natural numbers
3775:
3774:
3755:
3754:
3725:
3724:
3667:"is equal to" (
3661:
3660:
3659:
3658:
3657:
3649:
3648:
3637:
3607:
3606:
3583: and
3567:
3566:
3543:
3542:
3513:
3512:
3487:
3486:
3463:
3462:
3439:
3438:
3409:
3408:
3380:
3379:
3351:
3350:
3319:
3318:
3310:
3309:
3279:
3278:
3252: and
3236:
3235:
3204:
3203:
3172:
3161:
3160:
3138:
3137:
3109:
3108:
3083:
3082:
3054:
3053:
3022:
3021:
3014:quasi-reflexive
3013:
3012:
2982:
2981:
2953:
2952:
2921:
2920:
2912:
2911:
2881:
2880:
2852:
2851:
2820:
2819:
2811:
2810:
2778:
2777:
2741:
2740:
2715:
2714:
2706:
2705:
2698:
2697:
2690:
2689:
2664:
2663:
2660:
2634:
2633:
2614:
2613:
2594:
2593:
2572:
2571:
2549:
2548:
2526:
2525:
2506:
2505:
2433:
2422:
2421:
2420:It is equal to
2399:
2398:
2379:
2378:
2359:
2358:
2339:
2338:
2308:
2307:
2285:
2284:
2261:
2260:
2241:
2240:
2216:
2205:
2204:
2185:
2184:
2148:
2143:
2142:
2123:
2122:
2121:, the relation
2103:
2102:
2035:
2030:
2029:
1989:
1988:
1963:
1962:
1939:
1938:
1919:
1918:
1915:
1860:
1859:
1836:
1835:
1813:
1812:
1810:binary relation
1796:
1769:
1768:
1743:
1742:
1717:
1716:
1682:
1681:
1658:
1657:
1651:
1643:
1639:
1630:
1626:
1604:
1603:
1586:
1585:
1564:
1563:
1532:
1531:
1505:
1504:
1489:
1488:
1480:
1479:
1461:
1460:
1445:
1444:
1436:
1435:
1417:
1416:
1401:
1400:
1392:
1391:
1376:
1375:
1360:
1359:
1348:
1331:
1330:
1322:
1305:
1304:
1289:
1288:
1283:
1269:
1268:
1257:
1254: and
1234:
1233:
1218:
1217:
1206:
1198:
1197:
1178:
1177:
1148:
1147:
1122:
1121:
1064:
1060:
1054:
1050:
1024:
1020:
1014:
1010:
992:
988:
982:
978:
947:
943:
925:
921:
915:
911:
880:
876:
848:
844:
838:
834:
813:
809:
781:
777:
766:
762:
746:
742:
714:
710:
704:
700:
694:
690:
674:
670:
642:
638:
622:
618:
612:
608:
602:
598:
570:
566:
550:
546:
523:
506:
502:
486:
482:
476:
472:
457:Prewellordering
439:
435:
414:
410:
404:
400:
372:
368:
347:
343:
310:
306:
280:
276:
248:
244:
206:
203:
186:
182:
151:
147:
139:
131:
55:
22:
17:
12:
11:
5:
4646:
4645:
4642:
4634:
4633:
4628:
4618:
4617:
4614:
4613:
4593:
4592:External links
4590:
4588:
4587:
4582:
4569:
4551:
4546:
4533:
4528:
4511:
4506:
4493:
4487:
4474:
4465:
4459:
4445:
4443:
4440:
4437:
4436:
4424:
4412:
4400:
4388:
4375:
4350:
4339:
4326:
4325:
4323:
4320:
4303:
4300:
4275:
4274:
4269:
4264:
4259:
4254:
4249:
4244:
4239:
4234:
4229:
4223:
4222:
4201:
4194:
4188:
4163:
4156:
4154:
4152:
4149:
4146:
4144:
4141:
4135:
4134:
4131:
4128:
4125:
4122:
4119:
4114:
4109:
4104:
4099:
4095:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4063:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4041:
4038:
4035:
4031:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
3999:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3967:
3966:
3961:
3956:
3954:Total preorder
3951:
3946:
3941:
3936:
3931:
3926:
3921:
3897:
3892:
3889:
3884:
3880:
3875:
3854:
3842:
3839:
3813:
3782:
3762:
3738:
3735:
3732:
3721:
3720:
3719:"is less than"
3717:
3714:
3707:
3700:
3693:
3692:
3689:
3686:
3679:
3672:
3651:
3650:
3642:
3641:
3640:
3639:
3638:
3636:
3633:
3620:
3617:
3614:
3594:
3591:
3588:
3580:
3577:
3574:
3564:
3563:antitransitive
3550:
3539:antitransitive
3526:
3523:
3520:
3500:
3497:
3494:
3484:
3470:
3446:
3435:
3434:
3433:
3432:
3429:anti-symmetric
3416:
3396:
3393:
3390:
3387:
3367:
3364:
3361:
3358:
3349:are such that
3338:
3335:
3332:
3329:
3326:
3315:
3306:
3295:
3292:
3289:
3286:
3266:
3263:
3260:
3257:
3249:
3246:
3243:
3234:are such that
3223:
3220:
3217:
3214:
3211:
3200:
3193:
3179:
3175:
3171:
3168:
3145:
3125:
3122:
3119:
3116:
3096:
3093:
3090:
3070:
3067:
3064:
3061:
3052:are such that
3041:
3038:
3035:
3032:
3029:
3018:
3009:
2998:
2995:
2992:
2989:
2969:
2966:
2963:
2960:
2951:are such that
2940:
2937:
2934:
2931:
2928:
2917:
2908:
2897:
2894:
2891:
2888:
2868:
2865:
2862:
2859:
2850:are such that
2839:
2836:
2833:
2830:
2827:
2816:
2807:
2791:
2788:
2785:
2770:if and only if
2757:
2754:
2751:
2748:
2728:
2725:
2722:
2711:
2699:anti-reflexive
2671:
2659:
2656:
2644:
2641:
2621:
2601:
2580:
2556:
2536:
2533:
2513:
2493:
2490:
2487:
2484:
2481:
2475:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2440:
2436:
2432:
2429:
2409:
2406:
2386:
2377:) relation on
2366:
2346:
2336:
2332:
2315:
2295:
2292:
2268:
2248:
2228:
2223:
2219:
2215:
2212:
2192:
2182:
2163:
2160:
2155:
2151:
2130:
2110:
2086:
2083:
2080:
2077:
2071:
2065:
2062:
2059:
2056:
2053:
2050:
2047:
2042:
2038:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1976:
1973:
1970:
1960:
1957:is said to be
1946:
1926:
1914:
1911:
1897:. Along with
1867:
1843:
1820:
1800:
1799:
1785:
1782:
1779:
1776:
1756:
1753:
1750:
1730:
1727:
1724:
1704:
1701:
1698:
1695:
1692:
1689:
1665:
1622:
1621:
1618:
1617:
1602:
1599:
1596:
1588:
1587:
1584:
1581:
1578:
1575:
1572:
1571:
1561:
1550:
1547:
1544:
1529:
1518:
1515:
1512:
1502:
1482:
1481:
1478:
1475:
1472:
1469:
1468:
1458:
1438:
1437:
1434:
1431:
1428:
1425:
1424:
1414:
1394:
1393:
1390:
1387:
1384:
1383:
1373:
1358:
1355:
1352:
1349:
1345: or
1342:
1339:
1336:
1333:
1332:
1329:
1326:
1323:
1319:
1316:
1313:
1312:
1302:
1287:
1284:
1280:
1277:
1274:
1271:
1270:
1267:
1264:
1261:
1258:
1251:
1248:
1245:
1242:
1241:
1231:
1216:
1213:
1210:
1207:
1203:
1200:
1199:
1196:
1193:
1190:
1187:
1185:
1175:
1164:
1161:
1158:
1155:
1135:
1132:
1129:
1117:
1116:
1111:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1068:
1067:
1057:
1047:
1042:
1037:
1032:
1027:
1017:
1007:
1002:
996:
995:
985:
975:
970:
965:
960:
955:
950:
940:
935:
929:
928:
918:
908:
903:
898:
893:
888:
883:
873:
868:
862:
861:
856:
851:
841:
831:
826:
821:
816:
806:
801:
795:
794:
789:
784:
774:
769:
759:
754:
749:
739:
734:
728:
727:
722:
717:
707:
697:
687:
682:
677:
667:
662:
656:
655:
650:
645:
635:
630:
625:
615:
605:
595:
590:
584:
583:
578:
573:
563:
558:
553:
543:
538:
533:
528:
520:
519:
514:
509:
499:
494:
489:
479:
469:
464:
459:
453:
452:
447:
442:
432:
427:
422:
417:
407:
397:
392:
386:
385:
380:
375:
365:
360:
355:
350:
340:
335:
330:
328:Total preorder
324:
323:
318:
313:
303:
298:
293:
288:
283:
273:
268:
262:
261:
256:
251:
241:
236:
231:
226:
221:
216:
211:
200:
199:
194:
189:
179:
174:
169:
164:
159:
154:
144:
136:
135:
133:
128:
126:
124:
122:
120:
117:
115:
113:
110:
109:
104:
99:
94:
89:
84:
79:
74:
69:
64:
57:
56:
54:
53:
46:
39:
31:
15:
13:
10:
9:
6:
4:
3:
2:
4644:
4643:
4632:
4629:
4627:
4624:
4623:
4621:
4610:
4606:
4605:
4600:
4599:"Reflexivity"
4596:
4595:
4591:
4585:
4579:
4575:
4570:
4562:
4561:
4556:
4552:
4549:
4547:0-674-55451-5
4543:
4539:
4534:
4531:
4529:0-387-98290-6
4525:
4521:
4517:
4512:
4509:
4507:0-486-42079-5
4503:
4499:
4494:
4490:
4488:1-133-05000-X
4484:
4481:. Wadsworth.
4480:
4475:
4471:
4466:
4462:
4460:0-7618-0922-8
4456:
4452:
4447:
4446:
4441:
4434:, p. 187
4433:
4428:
4425:
4421:
4416:
4413:
4410:
4404:
4401:
4398:, p. 337
4397:
4392:
4389:
4385:
4379:
4376:
4372:
4368:
4364:
4360:
4354:
4351:
4348:
4343:
4340:
4336:
4331:
4328:
4321:
4319:
4317:
4313:
4309:
4301:
4299:
4297:
4291:
4287:
4283:
4273:
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4253:
4250:
4248:
4245:
4243:
4240:
4238:
4235:
4233:
4230:
4228:
4225:
4224:
4219:
4215:
4211:
4204:
4195:
4192:
4189:
4185:
4181:
4177:
4173:
4166:
4157:
4155:
4153:
4150:
4147:
4145:
4142:
4140:
4137:
4136:
4132:
4129:
4126:
4123:
4120:
4115:
4110:
4105:
4100:
4097:
4096:
4092:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4064:
4060:
4057:
4054:
4051:
4048:
4045:
4042:
4039:
4036:
4033:
4032:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
4000:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3968:
3965:
3962:
3960:
3957:
3955:
3952:
3950:
3949:Partial order
3947:
3945:
3942:
3940:
3937:
3935:
3932:
3930:
3927:
3925:
3922:
3919:
3918:
3914:
3908:
3895:
3890:
3887:
3882:
3878:
3873:
3852:
3840:
3838:
3835:
3830:
3828:
3811:
3802:
3800:
3796:
3780:
3760:
3752:
3736:
3733:
3730:
3718:
3715:
3712:
3711:proper subset
3708:
3705:
3701:
3698:
3697:
3696:
3690:
3687:
3684:
3680:
3677:
3673:
3670:
3666:
3665:
3664:
3655:
3646:
3634:
3632:
3618:
3615:
3612:
3592:
3589:
3586:
3578:
3575:
3572:
3562:
3548:
3540:
3524:
3521:
3518:
3498:
3495:
3492:
3482:
3468:
3460:
3444:
3430:
3414:
3394:
3391:
3388:
3385:
3365:
3362:
3359:
3356:
3336:
3333:
3330:
3327:
3324:
3316:
3314:
3307:
3293:
3290:
3287:
3284:
3264:
3261:
3258:
3255:
3247:
3244:
3241:
3221:
3218:
3215:
3212:
3209:
3201:
3199:
3198:
3197:antisymmetric
3194:
3173:
3169:
3166:
3159:
3143:
3123:
3120:
3117:
3114:
3094:
3091:
3088:
3068:
3065:
3062:
3059:
3039:
3036:
3033:
3030:
3027:
3019:
3017:
3010:
2996:
2993:
2990:
2987:
2967:
2964:
2961:
2958:
2938:
2935:
2932:
2929:
2926:
2918:
2916:
2909:
2895:
2892:
2889:
2886:
2866:
2863:
2860:
2857:
2837:
2834:
2831:
2828:
2825:
2817:
2815:
2808:
2805:
2789:
2786:
2783:
2775:
2771:
2755:
2752:
2749:
2746:
2739:holds for no
2726:
2723:
2720:
2712:
2710:
2702:
2694:
2687:
2686:
2685:
2684:
2683:
2669:
2657:
2655:
2642:
2639:
2619:
2599:
2570:
2554:
2534:
2531:
2511:
2491:
2485:
2482:
2479:
2473:
2467:
2464:
2458:
2455:
2452:
2443:
2438:
2427:
2407:
2404:
2384:
2364:
2344:
2334:
2330:
2327:
2313:
2293:
2290:
2282:
2266:
2246:
2226:
2221:
2213:
2210:
2203:is the union
2190:
2181:
2178:
2175:
2161:
2158:
2153:
2128:
2108:
2100:
2081:
2078:
2075:
2069:
2060:
2057:
2054:
2045:
2040:
2026:
2012:
2009:
2003:
2000:
1997:
1974:
1971:
1968:
1961:if for every
1958:
1944:
1924:
1912:
1910:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1879:
1865:
1857:
1841:
1834:
1818:
1811:
1807:
1798:
1783:
1780:
1777:
1774:
1754:
1751:
1748:
1728:
1725:
1722:
1702:
1699:
1696:
1693:
1690:
1687:
1679:
1663:
1656:
1624:
1623:
1600:
1597:
1594:
1579:
1576:
1573:
1562:
1548:
1545:
1542:
1530:
1516:
1513:
1510:
1503:
1476:
1473:
1470:
1459:
1432:
1429:
1426:
1415:
1388:
1374:
1356:
1353:
1350:
1340:
1337:
1334:
1324:
1317:
1314:
1303:
1285:
1278:
1275:
1265:
1262:
1259:
1249:
1246:
1243:
1232:
1214:
1211:
1208:
1194:
1191:
1188:
1176:
1162:
1156:
1153:
1133:
1130:
1127:
1119:
1118:
1115:
1112:
1110:
1107:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1079:Antisymmetric
1077:
1075:
1072:
1070:
1069:
1058:
1048:
1043:
1038:
1033:
1028:
1018:
1008:
1003:
1001:
998:
997:
986:
976:
971:
966:
961:
956:
951:
941:
936:
934:
931:
930:
919:
909:
904:
899:
894:
889:
884:
874:
869:
867:
864:
863:
857:
852:
842:
832:
827:
822:
817:
807:
802:
800:
797:
796:
790:
785:
775:
770:
760:
755:
750:
740:
735:
733:
730:
729:
723:
718:
708:
698:
688:
683:
678:
668:
663:
661:
658:
657:
651:
646:
636:
631:
626:
616:
606:
596:
591:
589:
588:Well-ordering
586:
585:
579:
574:
564:
559:
554:
544:
539:
534:
529:
526:
522:
521:
515:
510:
500:
495:
490:
480:
470:
465:
460:
458:
455:
454:
448:
443:
433:
428:
423:
418:
408:
398:
393:
391:
388:
387:
381:
376:
366:
361:
356:
351:
341:
336:
331:
329:
326:
325:
319:
314:
304:
299:
294:
289:
284:
274:
269:
267:
266:Partial order
264:
263:
257:
252:
242:
237:
232:
227:
222:
217:
212:
209:
202:
201:
195:
190:
180:
175:
170:
165:
160:
155:
145:
142:
138:
137:
134:
129:
127:
125:
123:
121:
118:
116:
114:
112:
111:
108:
105:
103:
100:
98:
95:
93:
90:
88:
85:
83:
80:
78:
75:
73:
72:Antisymmetric
70:
68:
65:
63:
62:
59:
58:
52:
47:
45:
40:
38:
33:
29:
25:
21:
20:
4602:
4573:
4559:
4537:
4515:
4497:
4478:
4469:
4450:
4427:
4415:
4403:
4391:
4378:
4370:
4366:
4363:Russell 1920
4353:
4347:Schmidt 2010
4342:
4337:, p. 74
4330:
4315:
4311:
4305:
4289:
4285:
4281:
4278:
4217:
4213:
4209:
4202:
4190:
4183:
4179:
4175:
4171:
4164:
4138:
3933:
3912:
3844:
3831:
3803:
3795:even numbers
3751:real numbers
3722:
3694:
3683:divisibility
3662:
3605:implies not
3511:implies not
3436:
3317:if whenever
3308:
3202:if whenever
3195:
3011:
2919:if whenever
2910:
2818:if whenever
2809:
2707:aliorelative
2704:
2696:
2688:
2661:
2328:
2176:
2027:
1916:
1903:transitivity
1894:
1890:
1887:real numbers
1880:
1878:to itself.
1855:
1803:
1652:
1103:
1089:Well-founded
207:(Quasiorder)
96:
82:Well-founded
4306:Authors in
3959:Total order
3681:"divides" (
3407:A relation
3311:coreflexive
2691:irreflexive
2682:is called:
2306:A relation
2097:denote the
1937:on the set
1917:A relation
1913:Definitions
1895:reflexivity
1883:is equal to
1806:mathematics
1109:Irreflexive
390:Total order
102:Irreflexive
4620:Categories
4442:References
4359:C S Peirce
4294:refers to
4279:Note that
3929:Transitive
3911:Number of
3483:asymmetric
3481:is called
3459:asymmetric
2774:complement
2279:that is a
1680:: for all
1678:transitive
1114:Asymmetric
107:Asymmetric
24:Transitive
4609:EMS Press
4335:Levy 1979
4316:reflexive
3939:Symmetric
3934:Reflexive
3888:−
3749:) on the
3334:∈
3219:∈
3170:∪
3037:∈
2936:∈
2835:∈
2787:×
2750:∈
2620:≤
2600:≤
2483:≠
2465:∈
2431:∖
2365:⊆
2247:⊆
2214:∪
2159:⊆
2079:∈
2010:∈
1972:∈
1959:reflexive
1856:reflexive
1591:not
1583:⇒
1539:not
1474:∧
1430:∨
1328:⇒
1318:≠
1273:⇒
1202:⇒
1160:∅
1157:≠
1104:Reflexive
1099:Has meets
1094:Has joins
1084:Connected
1074:Symmetric
205:Preorder
132:reflexive
97:Reflexive
92:Has meets
87:Has joins
77:Connected
67:Symmetric
4557:(1920).
3944:Preorder
3834:integers
3669:equality
3635:Examples
2281:superset
1899:symmetry
1648:✗
1635:✗
1045:✗
1040:✗
1035:✗
1030:✗
1005:✗
973:✗
968:✗
963:✗
958:✗
953:✗
938:✗
906:✗
901:✗
896:✗
891:✗
886:✗
871:✗
859:✗
854:✗
829:✗
824:✗
819:✗
804:✗
792:✗
787:✗
772:✗
757:✗
752:✗
737:✗
725:✗
720:✗
685:✗
680:✗
665:✗
653:✗
648:✗
633:✗
628:✗
593:✗
581:✗
576:✗
561:✗
556:✗
541:✗
536:✗
531:✗
517:✗
512:✗
497:✗
492:✗
467:✗
462:✗
450:✗
445:✗
430:✗
425:✗
420:✗
395:✗
383:✗
378:✗
363:✗
358:✗
353:✗
338:✗
333:✗
321:✗
316:✗
301:✗
296:✗
291:✗
286:✗
271:✗
259:✗
254:✗
239:✗
234:✗
229:✗
224:✗
219:✗
214:✗
197:✗
192:✗
177:✗
172:✗
167:✗
162:✗
157:✗
4611:, 2001
4409:A053763
4272:A000110
4267:A000142
4262:A000670
4257:A001035
4252:A000798
4247:A006125
4242:A053763
4237:A006905
4232:A002416
3704:coprime
3537:), nor
2567:on the
660:Lattice
4580:
4544:
4526:
4504:
4485:
4457:
4361:; see
4102:65,536
3709:"is a
3676:subset
3674:"is a
2477:
2471:
2073:
2067:
1485:exists
1441:exists
1397:exists
26:
4564:(PDF)
4322:Notes
4117:1,024
4112:4,096
4107:3,994
2569:reals
1831:on a
1767:then
130:Anti-
4578:ISBN
4542:ISBN
4524:ISBN
4502:ISBN
4483:ISBN
4455:ISBN
4382:The
4227:OEIS
3773:and
3734:>
3702:"is
3107:and
2772:its
2640:<
2555:<
2329:The
2177:The
1901:and
1808:, a
1741:and
1146:and
4369:or
4133:15
4124:219
4121:355
4072:171
4069:512
3924:Any
3801:.
3713:of"
3631:).
3565:if
3561:is
3485:if
2776:in
2703:or
2632:is
2337:of
2333:or
2283:of
2183:of
2101:on
1854:is
1833:set
1804:In
1715:if
1676:be
1386:min
4622::
4607:,
4601:,
4518:,
4318:.
4298:.
4288:,
4216:,
4205:=0
4193:!
4182:,
4167:=0
4151:2
4148:2
4143:2
4130:24
4127:75
4093:5
4087:13
4084:19
4081:29
4078:64
4075:64
4061:2
4040:13
4037:16
4029:1
3997:1
2695:,
2174:.
2046::=
2025:.
1987:,
1909:.
4491:.
4463:.
4373:.
4292:)
4290:k
4286:n
4284:(
4282:S
4220:)
4218:k
4214:n
4212:(
4210:S
4203:k
4197:∑
4191:n
4186:)
4184:k
4180:n
4178:(
4176:S
4174:!
4172:k
4165:k
4159:∑
4139:n
4098:4
4090:6
4066:3
4058:2
4055:3
4052:3
4049:4
4046:8
4043:4
4034:2
4026:1
4023:1
4020:1
4017:1
4014:2
4011:1
4008:2
4005:2
4002:1
3994:1
3991:1
3988:1
3985:1
3982:1
3979:1
3976:1
3973:1
3970:0
3913:n
3896:.
3891:n
3883:2
3879:n
3874:2
3853:n
3812:R
3781:y
3761:x
3737:y
3731:x
3685:)
3671:)
3619:z
3616:R
3613:x
3593:z
3590:R
3587:y
3579:y
3576:R
3573:x
3549:R
3541:(
3525:x
3522:R
3519:y
3499:y
3496:R
3493:x
3469:R
3461:(
3445:X
3431:.
3415:R
3395:.
3392:y
3389:=
3386:x
3366:,
3363:y
3360:R
3357:x
3337:X
3331:y
3328:,
3325:x
3294:.
3291:y
3288:=
3285:x
3265:,
3262:x
3259:R
3256:y
3248:y
3245:R
3242:x
3222:X
3216:y
3213:,
3210:x
3178:T
3174:R
3167:R
3144:R
3124:.
3121:y
3118:R
3115:y
3095:x
3092:R
3089:x
3069:,
3066:y
3063:R
3060:x
3040:X
3034:y
3031:,
3028:x
2997:.
2994:y
2991:R
2988:y
2968:,
2965:y
2962:R
2959:x
2939:X
2933:y
2930:,
2927:x
2896:.
2893:x
2890:R
2887:x
2867:,
2864:y
2861:R
2858:x
2838:X
2832:y
2829:,
2826:x
2790:X
2784:X
2756:.
2753:X
2747:x
2727:x
2724:R
2721:x
2670:R
2643:.
2579:R
2535:.
2532:R
2512:R
2492:.
2489:}
2486:y
2480:x
2474::
2468:R
2462:)
2459:y
2456:,
2453:x
2450:(
2447:{
2444:=
2439:X
2435:I
2428:R
2408:.
2405:R
2385:X
2345:R
2314:R
2294:.
2291:R
2267:X
2227:,
2222:X
2218:I
2211:R
2191:R
2162:R
2154:X
2150:I
2129:R
2109:X
2085:}
2082:X
2076:x
2070::
2064:)
2061:x
2058:,
2055:x
2052:(
2049:{
2041:X
2037:I
2013:R
2007:)
2004:x
2001:,
1998:x
1995:(
1975:X
1969:x
1945:X
1925:R
1866:X
1842:X
1819:R
1784:.
1781:c
1778:R
1775:a
1755:c
1752:R
1749:b
1729:b
1726:R
1723:a
1703:,
1700:c
1697:,
1694:b
1691:,
1688:a
1664:R
1644:Y
1631:Y
1601:a
1598:R
1595:b
1580:b
1577:R
1574:a
1549:a
1546:R
1543:a
1517:a
1514:R
1511:a
1477:b
1471:a
1433:b
1427:a
1389:S
1357:a
1354:R
1351:b
1341:b
1338:R
1335:a
1325:b
1315:a
1286:b
1279:=
1276:a
1266:a
1263:R
1260:b
1250:b
1247:R
1244:a
1215:a
1212:R
1209:b
1195:b
1192:R
1189:a
1163::
1154:S
1134:b
1131:,
1128:a
1065:Y
1055:Y
1025:Y
1015:Y
993:Y
983:Y
948:Y
926:Y
916:Y
881:Y
849:Y
839:Y
814:Y
782:Y
767:Y
747:Y
715:Y
705:Y
695:Y
675:Y
643:Y
623:Y
613:Y
603:Y
571:Y
551:Y
507:Y
487:Y
477:Y
440:Y
415:Y
405:Y
373:Y
348:Y
311:Y
281:Y
249:Y
187:Y
152:Y
50:e
43:t
36:v
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