1651:
1638:
1072:
1062:
1032:
1022:
1000:
990:
955:
933:
923:
888:
856:
846:
821:
789:
774:
754:
722:
712:
702:
682:
650:
630:
620:
610:
578:
558:
514:
494:
484:
447:
422:
412:
380:
355:
318:
288:
256:
194:
159:
3847:
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
3835:
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
3656:
3764:. 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
3665:
1382:
1311:
1626:
1240:
2513:
1511:
1467:
1423:
2106:
1648:
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
1581:
1478:
1434:
1393:
1322:
1251:
1195:
2183:
2248:
3201:
3286:
3614:
1184:
1570:
3917:
2601:
2386:
2268:
2034:
2811:
1317:
3358:
3243:
3061:
2960:
2859:
2777:
3758:
1996:
1724:
1246:
2664:
2641:
2621:
3416:
3387:
3315:
3145:
3090:
3018:
2989:
2917:
2888:
2576:
1805:
3640:
3546:
3520:
3116:
2748:
1776:
1750:
1538:
1155:
2556:
2429:
2315:
3874:
3833:
3802:
3782:
3570:
3490:
3466:
3436:
3165:
2691:
2533:
2406:
2366:
2335:
2288:
2212:
2150:
2130:
1966:
1946:
1887:
1863:
1840:
1685:
59:
1576:
1190:
4636:
2434:
4479:
Fonseca de
Oliveira, José Nuno; Pereira Cunha Rodrigues, César de Jesus (2004), "Transposing relations: from Maybe functions to hash tables",
1473:
4592:
1429:
1388:
4306:
3147:
Equivalently, a binary relation is quasi-reflexive if and only if it is both left quasi-reflexive and right quasi-reflexive. A relation
4530:
4556:
4538:
4516:
4497:
4469:
3734:
An example of an irreflexive relation, which means that it does not relate any element to itself, is the "greater than" relation (
2042:
3840:, which is always left quasi-reflexive but not necessarily right quasi-reflexive, and thus not necessarily quasi-reflexive.
52:
4619:
4641:
4394:
2155:
4614:
2217:
3173:
45:
3248:
3679:
3579:
3439:
3207:
3031:
if every element that is part of some relation is related to itself. Explicitly, this means that whenever
2784:
1893:
1089:
82:
1160:
3959:
1099:
92:
1544:
3974:
3964:
1917:
1665:
876:
151:
4318:
3939:
3469:
2814:
1913:
1688:
1124:
535:
117:
34:
3879:
2584:
1377:{\displaystyle {\begin{aligned}a\neq {}&b\Rightarrow \\aRb{\text{ or }}&bRa\end{aligned}}}
4609:
3949:
3837:
1909:
1094:
1084:
1010:
87:
77:
2371:
2253:
2001:
2790:
1306:{\displaystyle {\begin{aligned}aRb{\text{ and }}&bRa\\\Rightarrow a={}&b\end{aligned}}}
4588:
4552:
4534:
4512:
4493:
4465:
3331:
3216:
3168:
3034:
2933:
2832:
2535:
can, in a sense, be seen as a construction that is the "opposite" of the reflexive closure of
2190:
2109:
1843:
3808:, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of
2753:
1644:
indicates that the column's property is always true the row's term (at the very left), while
4565:
3737:
2817:
is necessarily irreflexive. A transitive and irreflexive relation is necessarily asymmetric.
1975:
1694:
809:
742:
2646:
2626:
2606:
4321:
often use different terminology. Reflexive relations in the mathematical sense are called
3934:
3392:
3363:
3291:
3121:
3066:
2994:
2965:
2893:
2864:
2561:
1820:
1781:
670:
467:
38:
3619:
3525:
3499:
3095:
2727:
1808:
A term's definition may require additional properties that are not listed in this table.
1755:
1729:
1517:
1134:
2538:
2411:
2297:
1900:, since every real number is equal to itself. A reflexive relation is said to have the
17:
3859:
3818:
3809:
3787:
3767:
3555:
3549:
3475:
3451:
3421:
3150:
2780:
2676:
2518:
2391:
2351:
2320:
2273:
2197:
2135:
2115:
1951:
1931:
1872:
1848:
1825:
1670:
338:
4630:
3721:
1109:
1104:
943:
276:
102:
97:
3655:
4569:
3969:
3805:
3761:
2579:
1897:
1816:
400:
2673:
There are several definitions related to the reflexive property. The relation
1621:{\displaystyle {\begin{aligned}aRb\Rightarrow \\{\text{not }}bRa\end{aligned}}}
4369:
1235:{\displaystyle {\begin{aligned}&aRb\\\Rightarrow {}&bRa\end{aligned}}}
598:
4462:
Deductive Logic – An
Introduction to Evaluation Techniques and Logical Theory
2508:{\displaystyle R\setminus \operatorname {I} _{X}=\{(x,y)\in R~:~x\neq y\}.}
4406:
3954:
2291:
215:
3664:
1506:{\displaystyle {\begin{aligned}a\wedge b\\{\text{exists}}\end{aligned}}}
3844:
3714:
3693:
2558:
For example, the reflexive closure of the canonical strict inequality
1462:{\displaystyle {\begin{aligned}a\vee b\\{\text{exists}}\end{aligned}}}
3686:
1418:{\displaystyle {\begin{aligned}\min S\\{\text{exists}}\end{aligned}}}
2337:
is reflexive if and only if it is equal to its reflexive closure.
2250:
which can equivalently be defined as the smallest (with respect to
4325:
in philosophical logic, and quasi-reflexive relations are called
4237:
4551:, Revised Edition, Reprinted 2003, Harvard University Press,
4419:
4376:, p. 32. Russell also introduces two equivalent terms
4282:
4277:
4272:
4267:
4262:
4257:
4252:
4247:
4242:
2101:{\displaystyle \operatorname {I} _{X}:=\{(x,x)~:~x\in X\}}
2724:
if it does not relate any element to itself; that is, if
3843:
An example of a coreflexive relation is the relation on
3438:
is coreflexive if and only if its symmetric closure is
4577:(2nd ed.). London: George Allen & Unwin, Ltd.
4407:
Fonseca de
Oliveira & Pereira Cunha Rodrigues 2004
3882:
3862:
3821:
3790:
3770:
3740:
3622:
3582:
3558:
3528:
3502:
3478:
3454:
3424:
3395:
3366:
3334:
3294:
3251:
3219:
3176:
3153:
3124:
3098:
3069:
3037:
2997:
2968:
2936:
2896:
2867:
2835:
2793:
2756:
2730:
2679:
2649:
2629:
2609:
2587:
2564:
2541:
2521:
2437:
2414:
2394:
2374:
2354:
2323:
2300:
2276:
2256:
2220:
2200:
2158:
2138:
2118:
2045:
2004:
1978:
1954:
1934:
1875:
1851:
1828:
1784:
1758:
1732:
1697:
1673:
1579:
1547:
1520:
1476:
1432:
1391:
1320:
1249:
1193:
1163:
1137:
4488:
Hausman, Alan; Kahane, Howard; Tidman, Paul (2013).
1892:
An example of a reflexive relation is the relation "
27:
Binary relation that relates every element to itself
4430:
3911:
3868:
3827:
3796:
3776:
3752:
3634:
3608:
3564:
3540:
3514:
3484:
3460:
3430:
3410:
3381:
3352:
3309:
3280:
3237:
3195:
3159:
3139:
3110:
3084:
3055:
3012:
2983:
2954:
2911:
2882:
2853:
2805:
2771:
2742:
2685:
2658:
2635:
2615:
2595:
2570:
2550:
2527:
2507:
2423:
2400:
2380:
2360:
2329:
2309:
2282:
2262:
2242:
2206:
2177:
2144:
2124:
2100:
2028:
1990:
1960:
1940:
1916:, reflexivity is one of three properties defining
1881:
1857:
1834:
1799:
1770:
1744:
1718:
1679:
1620:
1564:
1532:
1505:
1461:
1417:
1376:
1305:
1234:
1178:
1149:
2178:{\displaystyle \operatorname {I} _{X}\subseteq R}
1396:
4511:, Perspectives in Mathematical Logic, Dover,
4442:
3926:-element binary relations of different types
2243:{\displaystyle R\cup \operatorname {I} _{X},}
1661:in the "Antisymmetric" column, respectively.
53:
8:
4490:Logic and Philosophy – A Modern Introduction
3196:{\displaystyle R\cup R^{\operatorname {T} }}
2499:
2457:
2095:
2059:
3706:Examples of irreflexive relations include:
4418:On-Line Encyclopedia of Integer Sequences
60:
46:
29:
3892:
3887:
3881:
3861:
3820:
3815:An example of a quasi-reflexive relation
3789:
3769:
3739:
3674:Examples of reflexive relations include:
3621:
3592:
3581:
3557:
3527:
3501:
3477:
3453:
3423:
3394:
3365:
3333:
3293:
3261:
3250:
3218:
3187:
3175:
3152:
3123:
3097:
3068:
3036:
2996:
2967:
2935:
2895:
2866:
2834:
2792:
2755:
2729:
2678:
2648:
2628:
2608:
2589:
2588:
2586:
2563:
2540:
2520:
2448:
2436:
2413:
2393:
2373:
2353:
2322:
2299:
2275:
2255:
2231:
2219:
2199:
2163:
2157:
2137:
2117:
2050:
2044:
2003:
1977:
1953:
1933:
1874:
1850:
1827:
1783:
1757:
1731:
1696:
1672:
1600:
1580:
1578:
1548:
1546:
1519:
1494:
1477:
1475:
1450:
1433:
1431:
1406:
1392:
1390:
1354:
1331:
1321:
1319:
1292:
1263:
1250:
1248:
1215:
1194:
1192:
1162:
1136:
3920:
3856:The number of reflexive relations on an
4571:Introduction to Mathematical Philosophy
4460:Clarke, D.S.; Behling, Richard (1998).
4373:
4357:
4338:
3448:A reflexive relation on a nonempty set
2441:
2408:that has the same reflexive closure as
1170:
43:
4397:calls this property quasi-reflexivity.
3167:is quasi-reflexive if and only if its
3281:{\displaystyle xRy{\text{ and }}yRx,}
7:
4579:(Online corrected edition, Feb 2010)
4345:
3804:is even" is reflexive on the set of
3609:{\displaystyle xRy{\text{ and }}yRz}
1664:All definitions tacitly require the
4481:Mathematics of Program Construction
4307:Stirling numbers of the second kind
3848:the same set is always transitive.
3203:is left (or right) quasi-reflexive.
2623:whereas the reflexive reduction of
2603:is the usual non-strict inequality
1179:{\displaystyle S\neq \varnothing :}
4531:Undergraduate Texts in Mathematics
3188:
2445:
2228:
2160:
2047:
25:
4431:Hausman, Kahane & Tidman 2013
3717:to" on the integers larger than 1
2368:is the smallest (with respect to
3663:
3654:
3468:can neither be irreflexive, nor
1649:
1636:
1565:{\displaystyle {\text{not }}aRa}
1119:
1070:
1060:
1030:
1020:
998:
988:
953:
931:
921:
886:
854:
844:
819:
787:
772:
752:
720:
710:
700:
680:
648:
628:
618:
608:
576:
556:
512:
492:
482:
445:
420:
410:
378:
353:
316:
286:
254:
192:
157:
112:
4464:. University Press of America.
1869:if it relates every element of
4637:Properties of binary relations
4587:, Cambridge University Press,
2472:
2460:
2074:
2062:
2017:
2005:
1657:in the "Symmetric" column and
1593:
1338:
1283:
1212:
1:
3852:Number of reflexive relations
3699:"is greater than or equal to"
1658:
1645:
1055:
1050:
1045:
1040:
1015:
983:
978:
973:
968:
963:
948:
916:
911:
906:
901:
896:
881:
869:
864:
839:
834:
829:
814:
802:
797:
782:
767:
762:
747:
735:
730:
695:
690:
675:
663:
658:
643:
638:
603:
591:
586:
571:
566:
551:
546:
541:
527:
522:
507:
502:
477:
472:
460:
455:
440:
435:
430:
405:
393:
388:
373:
368:
363:
348:
343:
331:
326:
311:
306:
301:
296:
281:
269:
264:
249:
244:
239:
234:
229:
224:
207:
202:
187:
182:
177:
172:
167:
3912:{\displaystyle 2^{n^{2}-n}.}
2596:{\displaystyle \mathbb {R} }
4615:Encyclopedia of Mathematics
4525:Lidl, R.; Pilz, G. (1998),
2515:The reflexive reduction of
4658:
3702:"is less than or equal to"
2779:A relation is irreflexive
2381:{\displaystyle \subseteq }
2263:{\displaystyle \subseteq }
2029:{\displaystyle (x,x)\in R}
4583:Schmidt, Gunther (2010),
4443:Clarke & Behling 1998
2806:{\displaystyle X\times X}
4527:Applied abstract algebra
3353:{\displaystyle x,y\in X}
3238:{\displaystyle x,y\in X}
3056:{\displaystyle x,y\in X}
2955:{\displaystyle x,y\in X}
2854:{\displaystyle x,y\in X}
2270:) reflexive relation on
18:Quasi-reflexive relation
4395:Encyclopedia Britannica
2772:{\displaystyle x\in X.}
4585:Relational Mathematics
3913:
3870:
3829:
3798:
3778:
3754:
3753:{\displaystyle x>y}
3636:
3610:
3566:
3542:
3516:
3486:
3462:
3432:
3412:
3383:
3354:
3311:
3282:
3239:
3197:
3161:
3141:
3112:
3086:
3057:
3014:
2985:
2956:
2913:
2884:
2855:
2807:
2773:
2744:
2687:
2660:
2637:
2617:
2597:
2572:
2552:
2529:
2509:
2425:
2402:
2382:
2362:
2331:
2311:
2284:
2264:
2244:
2208:
2179:
2146:
2126:
2102:
2039:Equivalently, letting
2030:
1992:
1991:{\displaystyle x\in X}
1962:
1942:
1904:or is said to possess
1883:
1859:
1836:
1801:
1772:
1746:
1720:
1719:{\displaystyle a,b,c,}
1681:
1622:
1566:
1534:
1507:
1463:
1419:
1378:
1307:
1236:
1180:
1151:
4547:Quine, W. V. (1951),
3914:
3871:
3830:
3799:
3779:
3755:
3637:
3611:
3567:
3543:
3517:
3487:
3463:
3433:
3413:
3384:
3355:
3312:
3283:
3240:
3198:
3162:
3142:
3113:
3087:
3058:
3015:
2986:
2957:
2924:right quasi-reflexive
2914:
2885:
2856:
2808:
2774:
2745:
2688:
2661:
2659:{\displaystyle <.}
2638:
2636:{\displaystyle \leq }
2618:
2616:{\displaystyle \leq }
2598:
2573:
2553:
2530:
2510:
2426:
2403:
2383:
2363:
2332:
2312:
2285:
2265:
2245:
2209:
2180:
2147:
2127:
2103:
2031:
1993:
1963:
1943:
1918:equivalence relations
1884:
1860:
1837:
1802:
1773:
1747:
1721:
1682:
1623:
1567:
1535:
1508:
1464:
1420:
1379:
1308:
1237:
1181:
1152:
1131:Definitions, for all
4368:This term is due to
3975:Equivalence relation
3880:
3860:
3819:
3788:
3768:
3738:
3620:
3580:
3556:
3526:
3500:
3476:
3452:
3422:
3411:{\displaystyle x=y.}
3393:
3382:{\displaystyle xRy,}
3364:
3332:
3310:{\displaystyle x=y.}
3292:
3249:
3217:
3174:
3151:
3140:{\displaystyle yRy.}
3122:
3096:
3085:{\displaystyle xRy,}
3067:
3035:
3013:{\displaystyle yRy.}
2995:
2984:{\displaystyle xRy,}
2966:
2934:
2912:{\displaystyle xRx.}
2894:
2883:{\displaystyle xRy,}
2865:
2833:
2823:left quasi-reflexive
2791:
2754:
2728:
2677:
2647:
2627:
2607:
2585:
2571:{\displaystyle <}
2562:
2539:
2519:
2435:
2412:
2392:
2372:
2352:
2321:
2298:
2274:
2254:
2218:
2198:
2156:
2136:
2116:
2043:
2002:
1976:
1952:
1932:
1873:
1849:
1826:
1800:{\displaystyle aRc.}
1782:
1756:
1730:
1695:
1671:
1666:homogeneous relation
1577:
1545:
1518:
1474:
1430:
1389:
1318:
1247:
1191:
1161:
1135:
877:Strict partial order
152:Equivalence relation
4642:Reflexive relations
4533:, Springer-Verlag,
4483:, Springer: 334–356
4319:philosophical logic
4313:Philosophical logic
3927:
3689:of" (set inclusion)
3635:{\displaystyle xRz}
3541:{\displaystyle yRx}
3515:{\displaystyle xRy}
3111:{\displaystyle xRx}
2815:asymmetric relation
2743:{\displaystyle xRx}
2669:Related definitions
2342:reflexive reduction
1771:{\displaystyle bRc}
1745:{\displaystyle aRb}
1533:{\displaystyle aRa}
1150:{\displaystyle a,b}
536:Well-quasi-ordering
4549:Mathematical Logic
4433:, pp. 327–328
4378:to be contained in
3921:
3909:
3866:
3838:Euclidean relation
3825:
3794:
3774:
3750:
3632:
3606:
3562:
3538:
3512:
3482:
3458:
3428:
3408:
3379:
3350:
3307:
3278:
3235:
3193:
3157:
3137:
3108:
3082:
3053:
3010:
2981:
2952:
2909:
2880:
2851:
2803:
2769:
2740:
2683:
2656:
2633:
2613:
2593:
2568:
2551:{\displaystyle R.}
2548:
2525:
2505:
2424:{\displaystyle R.}
2421:
2398:
2378:
2358:
2346:irreflexive kernel
2327:
2310:{\displaystyle R.}
2307:
2280:
2260:
2240:
2204:
2175:
2142:
2122:
2098:
2026:
1988:
1958:
1938:
1902:reflexive property
1879:
1855:
1832:
1797:
1768:
1742:
1716:
1677:
1618:
1616:
1562:
1530:
1503:
1501:
1459:
1457:
1415:
1413:
1374:
1372:
1303:
1301:
1232:
1230:
1176:
1147:
1011:Strict total order
4594:978-0-521-76268-7
4566:Russell, Bertrand
4507:Levy, A. (1979),
4323:totally reflexive
4288:
4287:
3869:{\displaystyle n}
3828:{\displaystyle R}
3797:{\displaystyle y}
3777:{\displaystyle x}
3727:"is greater than"
3710:"is not equal to"
3595:
3565:{\displaystyle R}
3485:{\displaystyle R}
3461:{\displaystyle X}
3431:{\displaystyle R}
3389:then necessarily
3288:then necessarily
3264:
3169:symmetric closure
3160:{\displaystyle R}
3092:then necessarily
2991:then necessarily
2890:then necessarily
2813:is reflexive. An
2686:{\displaystyle R}
2528:{\displaystyle R}
2489:
2483:
2401:{\displaystyle X}
2361:{\displaystyle R}
2330:{\displaystyle R}
2283:{\displaystyle X}
2207:{\displaystyle R}
2191:reflexive closure
2145:{\displaystyle R}
2125:{\displaystyle X}
2110:identity relation
2085:
2079:
1961:{\displaystyle X}
1941:{\displaystyle R}
1882:{\displaystyle X}
1858:{\displaystyle X}
1835:{\displaystyle R}
1813:
1812:
1680:{\displaystyle R}
1631:
1630:
1603:
1551:
1497:
1453:
1409:
1357:
1266:
944:Strict weak order
130:Total, Semiconnex
16:(Redirected from
4649:
4623:
4597:
4578:
4576:
4561:
4543:
4521:
4509:Basic Set Theory
4503:
4484:
4475:
4446:
4440:
4434:
4428:
4422:
4416:
4410:
4404:
4398:
4391:
4385:
4366:
4360:
4355:
4349:
4343:
4304:
4232:
4219:
4218:
4198:
4181:
4180:
4129:
4124:
4119:
4114:
3928:
3918:
3916:
3915:
3910:
3905:
3904:
3897:
3896:
3876:-element set is
3875:
3873:
3872:
3867:
3834:
3832:
3831:
3826:
3803:
3801:
3800:
3795:
3783:
3781:
3780:
3775:
3759:
3757:
3756:
3751:
3667:
3658:
3641:
3639:
3638:
3633:
3615:
3613:
3612:
3607:
3596:
3593:
3571:
3569:
3568:
3563:
3547:
3545:
3544:
3539:
3521:
3519:
3518:
3513:
3491:
3489:
3488:
3483:
3467:
3465:
3464:
3459:
3437:
3435:
3434:
3429:
3417:
3415:
3414:
3409:
3388:
3386:
3385:
3380:
3359:
3357:
3356:
3351:
3324:
3323:
3316:
3314:
3313:
3308:
3287:
3285:
3284:
3279:
3265:
3262:
3244:
3242:
3241:
3236:
3202:
3200:
3199:
3194:
3192:
3191:
3166:
3164:
3163:
3158:
3146:
3144:
3143:
3138:
3117:
3115:
3114:
3109:
3091:
3089:
3088:
3083:
3062:
3060:
3059:
3054:
3027:
3026:
3019:
3017:
3016:
3011:
2990:
2988:
2987:
2982:
2961:
2959:
2958:
2953:
2926:
2925:
2918:
2916:
2915:
2910:
2889:
2887:
2886:
2881:
2860:
2858:
2857:
2852:
2825:
2824:
2812:
2810:
2809:
2804:
2778:
2776:
2775:
2770:
2749:
2747:
2746:
2741:
2720:
2719:
2712:
2711:
2704:
2703:
2692:
2690:
2689:
2684:
2665:
2663:
2662:
2657:
2642:
2640:
2639:
2634:
2622:
2620:
2619:
2614:
2602:
2600:
2599:
2594:
2592:
2577:
2575:
2574:
2569:
2557:
2555:
2554:
2549:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2487:
2481:
2453:
2452:
2430:
2428:
2427:
2422:
2407:
2405:
2404:
2399:
2387:
2385:
2384:
2379:
2367:
2365:
2364:
2359:
2336:
2334:
2333:
2328:
2316:
2314:
2313:
2308:
2289:
2287:
2286:
2281:
2269:
2267:
2266:
2261:
2249:
2247:
2246:
2241:
2236:
2235:
2213:
2211:
2210:
2205:
2184:
2182:
2181:
2176:
2168:
2167:
2152:is reflexive if
2151:
2149:
2148:
2143:
2131:
2129:
2128:
2123:
2107:
2105:
2104:
2099:
2083:
2077:
2055:
2054:
2035:
2033:
2032:
2027:
1997:
1995:
1994:
1989:
1967:
1965:
1964:
1959:
1947:
1945:
1944:
1939:
1896:" on the set of
1888:
1886:
1885:
1880:
1864:
1862:
1861:
1856:
1841:
1839:
1838:
1833:
1806:
1804:
1803:
1798:
1777:
1775:
1774:
1769:
1751:
1749:
1748:
1743:
1725:
1723:
1722:
1717:
1686:
1684:
1683:
1678:
1660:
1656:
1653:
1652:
1647:
1643:
1640:
1639:
1627:
1625:
1624:
1619:
1617:
1604:
1601:
1571:
1569:
1568:
1563:
1552:
1549:
1539:
1537:
1536:
1531:
1512:
1510:
1509:
1504:
1502:
1498:
1495:
1468:
1466:
1465:
1460:
1458:
1454:
1451:
1424:
1422:
1421:
1416:
1414:
1410:
1407:
1383:
1381:
1380:
1375:
1373:
1358:
1355:
1332:
1312:
1310:
1309:
1304:
1302:
1293:
1267:
1264:
1241:
1239:
1238:
1233:
1231:
1216:
1197:
1185:
1183:
1182:
1177:
1156:
1154:
1153:
1148:
1077:
1074:
1073:
1067:
1064:
1063:
1057:
1052:
1047:
1042:
1037:
1034:
1033:
1027:
1024:
1023:
1017:
1005:
1002:
1001:
995:
992:
991:
985:
980:
975:
970:
965:
960:
957:
956:
950:
938:
935:
934:
928:
925:
924:
918:
913:
908:
903:
898:
893:
890:
889:
883:
871:
866:
861:
858:
857:
851:
848:
847:
841:
836:
831:
826:
823:
822:
816:
810:Meet-semilattice
804:
799:
794:
791:
790:
784:
779:
776:
775:
769:
764:
759:
756:
755:
749:
743:Join-semilattice
737:
732:
727:
724:
723:
717:
714:
713:
707:
704:
703:
697:
692:
687:
684:
683:
677:
665:
660:
655:
652:
651:
645:
640:
635:
632:
631:
625:
622:
621:
615:
612:
611:
605:
593:
588:
583:
580:
579:
573:
568:
563:
560:
559:
553:
548:
543:
538:
529:
524:
519:
516:
515:
509:
504:
499:
496:
495:
489:
486:
485:
479:
474:
462:
457:
452:
449:
448:
442:
437:
432:
427:
424:
423:
417:
414:
413:
407:
395:
390:
385:
382:
381:
375:
370:
365:
360:
357:
356:
350:
345:
333:
328:
323:
320:
319:
313:
308:
303:
298:
293:
290:
289:
283:
271:
266:
261:
258:
257:
251:
246:
241:
236:
231:
226:
221:
219:
209:
204:
199:
196:
195:
189:
184:
179:
174:
169:
164:
161:
160:
154:
72:
71:
62:
55:
48:
41:
39:binary relations
30:
21:
4657:
4656:
4652:
4651:
4650:
4648:
4647:
4646:
4627:
4626:
4608:
4605:
4600:
4595:
4582:
4574:
4564:
4559:
4546:
4541:
4524:
4519:
4506:
4500:
4487:
4478:
4472:
4459:
4455:
4450:
4449:
4441:
4437:
4429:
4425:
4417:
4413:
4405:
4401:
4392:
4388:
4382:imply diversity
4367:
4363:
4356:
4352:
4344:
4340:
4335:
4315:
4291:
4217:
4211:
4210:
4209:
4207:
4179:
4173:
4172:
4171:
4169:
4127:
4122:
4117:
4112:
3931:Elements
3888:
3883:
3878:
3877:
3858:
3857:
3854:
3817:
3816:
3810:natural numbers
3786:
3785:
3766:
3765:
3736:
3735:
3678:"is equal to" (
3672:
3671:
3670:
3669:
3668:
3660:
3659:
3648:
3618:
3617:
3594: and
3578:
3577:
3554:
3553:
3524:
3523:
3498:
3497:
3474:
3473:
3450:
3449:
3420:
3419:
3391:
3390:
3362:
3361:
3330:
3329:
3321:
3320:
3290:
3289:
3263: and
3247:
3246:
3215:
3214:
3183:
3172:
3171:
3149:
3148:
3120:
3119:
3094:
3093:
3065:
3064:
3033:
3032:
3025:quasi-reflexive
3024:
3023:
2993:
2992:
2964:
2963:
2932:
2931:
2923:
2922:
2892:
2891:
2863:
2862:
2831:
2830:
2822:
2821:
2789:
2788:
2752:
2751:
2726:
2725:
2717:
2716:
2709:
2708:
2701:
2700:
2675:
2674:
2671:
2645:
2644:
2625:
2624:
2605:
2604:
2583:
2582:
2560:
2559:
2537:
2536:
2517:
2516:
2444:
2433:
2432:
2431:It is equal to
2410:
2409:
2390:
2389:
2370:
2369:
2350:
2349:
2319:
2318:
2296:
2295:
2272:
2271:
2252:
2251:
2227:
2216:
2215:
2196:
2195:
2159:
2154:
2153:
2134:
2133:
2132:, the relation
2114:
2113:
2046:
2041:
2040:
2000:
1999:
1974:
1973:
1950:
1949:
1930:
1929:
1926:
1871:
1870:
1847:
1846:
1824:
1823:
1821:binary relation
1807:
1780:
1779:
1754:
1753:
1728:
1727:
1693:
1692:
1669:
1668:
1662:
1654:
1650:
1641:
1637:
1615:
1614:
1597:
1596:
1575:
1574:
1543:
1542:
1516:
1515:
1500:
1499:
1491:
1490:
1472:
1471:
1456:
1455:
1447:
1446:
1428:
1427:
1412:
1411:
1403:
1402:
1387:
1386:
1371:
1370:
1359:
1342:
1341:
1333:
1316:
1315:
1300:
1299:
1294:
1280:
1279:
1268:
1265: and
1245:
1244:
1229:
1228:
1217:
1209:
1208:
1189:
1188:
1159:
1158:
1133:
1132:
1075:
1071:
1065:
1061:
1035:
1031:
1025:
1021:
1003:
999:
993:
989:
958:
954:
936:
932:
926:
922:
891:
887:
859:
855:
849:
845:
824:
820:
792:
788:
777:
773:
757:
753:
725:
721:
715:
711:
705:
701:
685:
681:
653:
649:
633:
629:
623:
619:
613:
609:
581:
577:
561:
557:
534:
517:
513:
497:
493:
487:
483:
468:Prewellordering
450:
446:
425:
421:
415:
411:
383:
379:
358:
354:
321:
317:
291:
287:
259:
255:
217:
214:
197:
193:
162:
158:
150:
142:
66:
33:
28:
23:
22:
15:
12:
11:
5:
4655:
4653:
4645:
4644:
4639:
4629:
4628:
4625:
4624:
4604:
4603:External links
4601:
4599:
4598:
4593:
4580:
4562:
4557:
4544:
4539:
4522:
4517:
4504:
4498:
4485:
4476:
4470:
4456:
4454:
4451:
4448:
4447:
4435:
4423:
4411:
4399:
4386:
4361:
4350:
4337:
4336:
4334:
4331:
4314:
4311:
4286:
4285:
4280:
4275:
4270:
4265:
4260:
4255:
4250:
4245:
4240:
4234:
4233:
4212:
4205:
4199:
4174:
4167:
4165:
4163:
4160:
4157:
4155:
4152:
4146:
4145:
4142:
4139:
4136:
4133:
4130:
4125:
4120:
4115:
4110:
4106:
4105:
4102:
4099:
4096:
4093:
4090:
4087:
4084:
4081:
4078:
4074:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4042:
4041:
4038:
4035:
4032:
4029:
4026:
4023:
4020:
4017:
4014:
4010:
4009:
4006:
4003:
4000:
3997:
3994:
3991:
3988:
3985:
3982:
3978:
3977:
3972:
3967:
3965:Total preorder
3962:
3957:
3952:
3947:
3942:
3937:
3932:
3908:
3903:
3900:
3895:
3891:
3886:
3865:
3853:
3850:
3824:
3793:
3773:
3749:
3746:
3743:
3732:
3731:
3730:"is less than"
3728:
3725:
3718:
3711:
3704:
3703:
3700:
3697:
3690:
3683:
3662:
3661:
3653:
3652:
3651:
3650:
3649:
3647:
3644:
3631:
3628:
3625:
3605:
3602:
3599:
3591:
3588:
3585:
3575:
3574:antitransitive
3561:
3550:antitransitive
3537:
3534:
3531:
3511:
3508:
3505:
3495:
3481:
3457:
3446:
3445:
3444:
3443:
3440:anti-symmetric
3427:
3407:
3404:
3401:
3398:
3378:
3375:
3372:
3369:
3360:are such that
3349:
3346:
3343:
3340:
3337:
3326:
3317:
3306:
3303:
3300:
3297:
3277:
3274:
3271:
3268:
3260:
3257:
3254:
3245:are such that
3234:
3231:
3228:
3225:
3222:
3211:
3204:
3190:
3186:
3182:
3179:
3156:
3136:
3133:
3130:
3127:
3107:
3104:
3101:
3081:
3078:
3075:
3072:
3063:are such that
3052:
3049:
3046:
3043:
3040:
3029:
3020:
3009:
3006:
3003:
3000:
2980:
2977:
2974:
2971:
2962:are such that
2951:
2948:
2945:
2942:
2939:
2928:
2919:
2908:
2905:
2902:
2899:
2879:
2876:
2873:
2870:
2861:are such that
2850:
2847:
2844:
2841:
2838:
2827:
2818:
2802:
2799:
2796:
2781:if and only if
2768:
2765:
2762:
2759:
2739:
2736:
2733:
2722:
2710:anti-reflexive
2682:
2670:
2667:
2655:
2652:
2632:
2612:
2591:
2567:
2547:
2544:
2524:
2504:
2501:
2498:
2495:
2492:
2486:
2480:
2477:
2474:
2471:
2468:
2465:
2462:
2459:
2456:
2451:
2447:
2443:
2440:
2420:
2417:
2397:
2388:) relation on
2377:
2357:
2347:
2343:
2326:
2306:
2303:
2279:
2259:
2239:
2234:
2230:
2226:
2223:
2203:
2193:
2174:
2171:
2166:
2162:
2141:
2121:
2097:
2094:
2091:
2088:
2082:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2053:
2049:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
1987:
1984:
1981:
1971:
1968:is said to be
1957:
1937:
1925:
1922:
1908:. Along with
1878:
1854:
1831:
1811:
1810:
1796:
1793:
1790:
1787:
1767:
1764:
1761:
1741:
1738:
1735:
1715:
1712:
1709:
1706:
1703:
1700:
1676:
1633:
1632:
1629:
1628:
1613:
1610:
1607:
1599:
1598:
1595:
1592:
1589:
1586:
1583:
1582:
1572:
1561:
1558:
1555:
1540:
1529:
1526:
1523:
1513:
1493:
1492:
1489:
1486:
1483:
1480:
1479:
1469:
1449:
1448:
1445:
1442:
1439:
1436:
1435:
1425:
1405:
1404:
1401:
1398:
1395:
1394:
1384:
1369:
1366:
1363:
1360:
1356: or
1353:
1350:
1347:
1344:
1343:
1340:
1337:
1334:
1330:
1327:
1324:
1323:
1313:
1298:
1295:
1291:
1288:
1285:
1282:
1281:
1278:
1275:
1272:
1269:
1262:
1259:
1256:
1253:
1252:
1242:
1227:
1224:
1221:
1218:
1214:
1211:
1210:
1207:
1204:
1201:
1198:
1196:
1186:
1175:
1172:
1169:
1166:
1146:
1143:
1140:
1128:
1127:
1122:
1117:
1112:
1107:
1102:
1097:
1092:
1087:
1082:
1079:
1078:
1068:
1058:
1053:
1048:
1043:
1038:
1028:
1018:
1013:
1007:
1006:
996:
986:
981:
976:
971:
966:
961:
951:
946:
940:
939:
929:
919:
914:
909:
904:
899:
894:
884:
879:
873:
872:
867:
862:
852:
842:
837:
832:
827:
817:
812:
806:
805:
800:
795:
785:
780:
770:
765:
760:
750:
745:
739:
738:
733:
728:
718:
708:
698:
693:
688:
678:
673:
667:
666:
661:
656:
646:
641:
636:
626:
616:
606:
601:
595:
594:
589:
584:
574:
569:
564:
554:
549:
544:
539:
531:
530:
525:
520:
510:
505:
500:
490:
480:
475:
470:
464:
463:
458:
453:
443:
438:
433:
428:
418:
408:
403:
397:
396:
391:
386:
376:
371:
366:
361:
351:
346:
341:
339:Total preorder
335:
334:
329:
324:
314:
309:
304:
299:
294:
284:
279:
273:
272:
267:
262:
252:
247:
242:
237:
232:
227:
222:
211:
210:
205:
200:
190:
185:
180:
175:
170:
165:
155:
147:
146:
144:
139:
137:
135:
133:
131:
128:
126:
124:
121:
120:
115:
110:
105:
100:
95:
90:
85:
80:
75:
68:
67:
65:
64:
57:
50:
42:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4654:
4643:
4640:
4638:
4635:
4634:
4632:
4621:
4617:
4616:
4611:
4610:"Reflexivity"
4607:
4606:
4602:
4596:
4590:
4586:
4581:
4573:
4572:
4567:
4563:
4560:
4558:0-674-55451-5
4554:
4550:
4545:
4542:
4540:0-387-98290-6
4536:
4532:
4528:
4523:
4520:
4518:0-486-42079-5
4514:
4510:
4505:
4501:
4499:1-133-05000-X
4495:
4492:. Wadsworth.
4491:
4486:
4482:
4477:
4473:
4471:0-7618-0922-8
4467:
4463:
4458:
4457:
4452:
4445:, p. 187
4444:
4439:
4436:
4432:
4427:
4424:
4421:
4415:
4412:
4409:, p. 337
4408:
4403:
4400:
4396:
4390:
4387:
4383:
4379:
4375:
4371:
4365:
4362:
4359:
4354:
4351:
4347:
4342:
4339:
4332:
4330:
4328:
4324:
4320:
4312:
4310:
4308:
4302:
4298:
4294:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4249:
4246:
4244:
4241:
4239:
4236:
4235:
4230:
4226:
4222:
4215:
4206:
4203:
4200:
4196:
4192:
4188:
4184:
4177:
4168:
4166:
4164:
4161:
4158:
4156:
4153:
4151:
4148:
4147:
4143:
4140:
4137:
4134:
4131:
4126:
4121:
4116:
4111:
4108:
4107:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4075:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4044:
4043:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4011:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3979:
3976:
3973:
3971:
3968:
3966:
3963:
3961:
3960:Partial order
3958:
3956:
3953:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3930:
3929:
3925:
3919:
3906:
3901:
3898:
3893:
3889:
3884:
3863:
3851:
3849:
3846:
3841:
3839:
3822:
3813:
3811:
3807:
3791:
3771:
3763:
3747:
3744:
3741:
3729:
3726:
3723:
3722:proper subset
3719:
3716:
3712:
3709:
3708:
3707:
3701:
3698:
3695:
3691:
3688:
3684:
3681:
3677:
3676:
3675:
3666:
3657:
3645:
3643:
3629:
3626:
3623:
3603:
3600:
3597:
3589:
3586:
3583:
3573:
3559:
3551:
3535:
3532:
3529:
3509:
3506:
3503:
3493:
3479:
3471:
3455:
3441:
3425:
3405:
3402:
3399:
3396:
3376:
3373:
3370:
3367:
3347:
3344:
3341:
3338:
3335:
3327:
3325:
3318:
3304:
3301:
3298:
3295:
3275:
3272:
3269:
3266:
3258:
3255:
3252:
3232:
3229:
3226:
3223:
3220:
3212:
3210:
3209:
3208:antisymmetric
3205:
3184:
3180:
3177:
3170:
3154:
3134:
3131:
3128:
3125:
3105:
3102:
3099:
3079:
3076:
3073:
3070:
3050:
3047:
3044:
3041:
3038:
3030:
3028:
3021:
3007:
3004:
3001:
2998:
2978:
2975:
2972:
2969:
2949:
2946:
2943:
2940:
2937:
2929:
2927:
2920:
2906:
2903:
2900:
2897:
2877:
2874:
2871:
2868:
2848:
2845:
2842:
2839:
2836:
2828:
2826:
2819:
2816:
2800:
2797:
2794:
2786:
2782:
2766:
2763:
2760:
2757:
2750:holds for no
2737:
2734:
2731:
2723:
2721:
2713:
2705:
2698:
2697:
2696:
2695:
2694:
2680:
2668:
2666:
2653:
2650:
2630:
2610:
2581:
2565:
2545:
2542:
2522:
2502:
2496:
2493:
2490:
2484:
2478:
2475:
2469:
2466:
2463:
2454:
2449:
2438:
2418:
2415:
2395:
2375:
2355:
2345:
2341:
2338:
2324:
2304:
2301:
2293:
2277:
2257:
2237:
2232:
2224:
2221:
2214:is the union
2201:
2192:
2189:
2186:
2172:
2169:
2164:
2139:
2119:
2111:
2092:
2089:
2086:
2080:
2071:
2068:
2065:
2056:
2051:
2037:
2023:
2020:
2014:
2011:
2008:
1985:
1982:
1979:
1972:if for every
1969:
1955:
1935:
1923:
1921:
1919:
1915:
1911:
1907:
1903:
1899:
1895:
1890:
1876:
1868:
1852:
1845:
1829:
1822:
1818:
1809:
1794:
1791:
1788:
1785:
1765:
1762:
1759:
1739:
1736:
1733:
1713:
1710:
1707:
1704:
1701:
1698:
1690:
1674:
1667:
1635:
1634:
1611:
1608:
1605:
1590:
1587:
1584:
1573:
1559:
1556:
1553:
1541:
1527:
1524:
1521:
1514:
1487:
1484:
1481:
1470:
1443:
1440:
1437:
1426:
1399:
1385:
1367:
1364:
1361:
1351:
1348:
1345:
1335:
1328:
1325:
1314:
1296:
1289:
1286:
1276:
1273:
1270:
1260:
1257:
1254:
1243:
1225:
1222:
1219:
1205:
1202:
1199:
1187:
1173:
1167:
1164:
1144:
1141:
1138:
1130:
1129:
1126:
1123:
1121:
1118:
1116:
1113:
1111:
1108:
1106:
1103:
1101:
1098:
1096:
1093:
1091:
1090:Antisymmetric
1088:
1086:
1083:
1081:
1080:
1069:
1059:
1054:
1049:
1044:
1039:
1029:
1019:
1014:
1012:
1009:
1008:
997:
987:
982:
977:
972:
967:
962:
952:
947:
945:
942:
941:
930:
920:
915:
910:
905:
900:
895:
885:
880:
878:
875:
874:
868:
863:
853:
843:
838:
833:
828:
818:
813:
811:
808:
807:
801:
796:
786:
781:
771:
766:
761:
751:
746:
744:
741:
740:
734:
729:
719:
709:
699:
694:
689:
679:
674:
672:
669:
668:
662:
657:
647:
642:
637:
627:
617:
607:
602:
600:
599:Well-ordering
597:
596:
590:
585:
575:
570:
565:
555:
550:
545:
540:
537:
533:
532:
526:
521:
511:
506:
501:
491:
481:
476:
471:
469:
466:
465:
459:
454:
444:
439:
434:
429:
419:
409:
404:
402:
399:
398:
392:
387:
377:
372:
367:
362:
352:
347:
342:
340:
337:
336:
330:
325:
315:
310:
305:
300:
295:
285:
280:
278:
277:Partial order
275:
274:
268:
263:
253:
248:
243:
238:
233:
228:
223:
220:
213:
212:
206:
201:
191:
186:
181:
176:
171:
166:
156:
153:
149:
148:
145:
140:
138:
136:
134:
132:
129:
127:
125:
123:
122:
119:
116:
114:
111:
109:
106:
104:
101:
99:
96:
94:
91:
89:
86:
84:
83:Antisymmetric
81:
79:
76:
74:
73:
70:
69:
63:
58:
56:
51:
49:
44:
40:
36:
32:
31:
19:
4613:
4584:
4570:
4548:
4526:
4508:
4489:
4480:
4461:
4438:
4426:
4414:
4402:
4389:
4381:
4377:
4374:Russell 1920
4364:
4358:Schmidt 2010
4353:
4348:, p. 74
4341:
4326:
4322:
4316:
4300:
4296:
4292:
4289:
4228:
4224:
4220:
4213:
4201:
4194:
4190:
4186:
4182:
4175:
4149:
3944:
3923:
3855:
3842:
3814:
3806:even numbers
3762:real numbers
3733:
3705:
3694:divisibility
3673:
3616:implies not
3522:implies not
3447:
3328:if whenever
3319:
3213:if whenever
3206:
3022:
2930:if whenever
2921:
2829:if whenever
2820:
2718:aliorelative
2715:
2707:
2699:
2672:
2339:
2187:
2038:
1927:
1914:transitivity
1905:
1901:
1898:real numbers
1891:
1889:to itself.
1866:
1814:
1663:
1114:
1100:Well-founded
218:(Quasiorder)
107:
93:Well-founded
4317:Authors in
3970:Total order
3692:"divides" (
3418:A relation
3322:coreflexive
2702:irreflexive
2693:is called:
2317:A relation
2108:denote the
1948:on the set
1928:A relation
1924:Definitions
1906:reflexivity
1894:is equal to
1817:mathematics
1120:Irreflexive
401:Total order
113:Irreflexive
4631:Categories
4453:References
4370:C S Peirce
4305:refers to
4290:Note that
3940:Transitive
3922:Number of
3494:asymmetric
3492:is called
3470:asymmetric
2785:complement
2290:that is a
1691:: for all
1689:transitive
1125:Asymmetric
118:Asymmetric
35:Transitive
4620:EMS Press
4346:Levy 1979
4327:reflexive
3950:Symmetric
3945:Reflexive
3899:−
3760:) on the
3345:∈
3230:∈
3181:∪
3048:∈
2947:∈
2846:∈
2798:×
2761:∈
2631:≤
2611:≤
2494:≠
2476:∈
2442:∖
2376:⊆
2258:⊆
2225:∪
2170:⊆
2090:∈
2021:∈
1983:∈
1970:reflexive
1867:reflexive
1602:not
1594:⇒
1550:not
1485:∧
1441:∨
1339:⇒
1329:≠
1284:⇒
1213:⇒
1171:∅
1168:≠
1115:Reflexive
1110:Has meets
1105:Has joins
1095:Connected
1085:Symmetric
216:Preorder
143:reflexive
108:Reflexive
103:Has meets
98:Has joins
88:Connected
78:Symmetric
4568:(1920).
3955:Preorder
3845:integers
3680:equality
3646:Examples
2292:superset
1910:symmetry
1659:✗
1646:✗
1056:✗
1051:✗
1046:✗
1041:✗
1016:✗
984:✗
979:✗
974:✗
969:✗
964:✗
949:✗
917:✗
912:✗
907:✗
902:✗
897:✗
882:✗
870:✗
865:✗
840:✗
835:✗
830:✗
815:✗
803:✗
798:✗
783:✗
768:✗
763:✗
748:✗
736:✗
731:✗
696:✗
691:✗
676:✗
664:✗
659:✗
644:✗
639:✗
604:✗
592:✗
587:✗
572:✗
567:✗
552:✗
547:✗
542:✗
528:✗
523:✗
508:✗
503:✗
478:✗
473:✗
461:✗
456:✗
441:✗
436:✗
431:✗
406:✗
394:✗
389:✗
374:✗
369:✗
364:✗
349:✗
344:✗
332:✗
327:✗
312:✗
307:✗
302:✗
297:✗
282:✗
270:✗
265:✗
250:✗
245:✗
240:✗
235:✗
230:✗
225:✗
208:✗
203:✗
188:✗
183:✗
178:✗
173:✗
168:✗
4622:, 2001
4420:A053763
4283:A000110
4278:A000142
4273:A000670
4268:A001035
4263:A000798
4258:A006125
4253:A053763
4248:A006905
4243:A002416
3715:coprime
3548:), nor
2578:on the
671:Lattice
4591:
4555:
4537:
4515:
4496:
4468:
4372:; see
4113:65,536
3720:"is a
3687:subset
3685:"is a
2488:
2482:
2084:
2078:
1496:exists
1452:exists
1408:exists
37:
4575:(PDF)
4333:Notes
4128:1,024
4123:4,096
4118:3,994
2580:reals
1842:on a
1778:then
141:Anti-
4589:ISBN
4553:ISBN
4535:ISBN
4513:ISBN
4494:ISBN
4466:ISBN
4393:The
4238:OEIS
3784:and
3745:>
3713:"is
3118:and
2783:its
2651:<
2566:<
2340:The
2188:The
1912:and
1819:, a
1752:and
1157:and
4380:or
4144:15
4135:219
4132:355
4083:171
4080:512
3935:Any
3812:.
3724:of"
3642:).
3576:if
3572:is
3496:if
2787:in
2714:or
2643:is
2348:of
2344:or
2294:of
2194:of
2112:on
1865:is
1844:set
1815:In
1726:if
1687:be
1397:min
4633::
4618:,
4612:,
4529:,
4329:.
4309:.
4299:,
4227:,
4216:=0
4204:!
4193:,
4178:=0
4162:2
4159:2
4154:2
4141:24
4138:75
4104:5
4098:13
4095:19
4092:29
4089:64
4086:64
4072:2
4051:13
4048:16
4040:1
4008:1
2706:,
2185:.
2057::=
2036:.
1998:,
1920:.
4502:.
4474:.
4384:.
4303:)
4301:k
4297:n
4295:(
4293:S
4231:)
4229:k
4225:n
4223:(
4221:S
4214:k
4208:∑
4202:n
4197:)
4195:k
4191:n
4189:(
4187:S
4185:!
4183:k
4176:k
4170:∑
4150:n
4109:4
4101:6
4077:3
4069:2
4066:3
4063:3
4060:4
4057:8
4054:4
4045:2
4037:1
4034:1
4031:1
4028:1
4025:2
4022:1
4019:2
4016:2
4013:1
4005:1
4002:1
3999:1
3996:1
3993:1
3990:1
3987:1
3984:1
3981:0
3924:n
3907:.
3902:n
3894:2
3890:n
3885:2
3864:n
3823:R
3792:y
3772:x
3748:y
3742:x
3696:)
3682:)
3630:z
3627:R
3624:x
3604:z
3601:R
3598:y
3590:y
3587:R
3584:x
3560:R
3552:(
3536:x
3533:R
3530:y
3510:y
3507:R
3504:x
3480:R
3472:(
3456:X
3442:.
3426:R
3406:.
3403:y
3400:=
3397:x
3377:,
3374:y
3371:R
3368:x
3348:X
3342:y
3339:,
3336:x
3305:.
3302:y
3299:=
3296:x
3276:,
3273:x
3270:R
3267:y
3259:y
3256:R
3253:x
3233:X
3227:y
3224:,
3221:x
3189:T
3185:R
3178:R
3155:R
3135:.
3132:y
3129:R
3126:y
3106:x
3103:R
3100:x
3080:,
3077:y
3074:R
3071:x
3051:X
3045:y
3042:,
3039:x
3008:.
3005:y
3002:R
2999:y
2979:,
2976:y
2973:R
2970:x
2950:X
2944:y
2941:,
2938:x
2907:.
2904:x
2901:R
2898:x
2878:,
2875:y
2872:R
2869:x
2849:X
2843:y
2840:,
2837:x
2801:X
2795:X
2767:.
2764:X
2758:x
2738:x
2735:R
2732:x
2681:R
2654:.
2590:R
2546:.
2543:R
2523:R
2503:.
2500:}
2497:y
2491:x
2485::
2479:R
2473:)
2470:y
2467:,
2464:x
2461:(
2458:{
2455:=
2450:X
2446:I
2439:R
2419:.
2416:R
2396:X
2356:R
2325:R
2305:.
2302:R
2278:X
2238:,
2233:X
2229:I
2222:R
2202:R
2173:R
2165:X
2161:I
2140:R
2120:X
2096:}
2093:X
2087:x
2081::
2075:)
2072:x
2069:,
2066:x
2063:(
2060:{
2052:X
2048:I
2024:R
2018:)
2015:x
2012:,
2009:x
2006:(
1986:X
1980:x
1956:X
1936:R
1877:X
1853:X
1830:R
1795:.
1792:c
1789:R
1786:a
1766:c
1763:R
1760:b
1740:b
1737:R
1734:a
1714:,
1711:c
1708:,
1705:b
1702:,
1699:a
1675:R
1655:Y
1642:Y
1612:a
1609:R
1606:b
1591:b
1588:R
1585:a
1560:a
1557:R
1554:a
1528:a
1525:R
1522:a
1488:b
1482:a
1444:b
1438:a
1400:S
1368:a
1365:R
1362:b
1352:b
1349:R
1346:a
1336:b
1326:a
1297:b
1290:=
1287:a
1277:a
1274:R
1271:b
1261:b
1258:R
1255:a
1226:a
1223:R
1220:b
1206:b
1203:R
1200:a
1174::
1165:S
1145:b
1142:,
1139:a
1076:Y
1066:Y
1036:Y
1026:Y
1004:Y
994:Y
959:Y
937:Y
927:Y
892:Y
860:Y
850:Y
825:Y
793:Y
778:Y
758:Y
726:Y
716:Y
706:Y
686:Y
654:Y
634:Y
624:Y
614:Y
582:Y
562:Y
518:Y
498:Y
488:Y
451:Y
426:Y
416:Y
384:Y
359:Y
322:Y
292:Y
260:Y
198:Y
163:Y
61:e
54:t
47:v
20:)
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