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defined on the power set of a set (satisfying certain conditions, one of which was that the set itself was closed) but this had to be modified to fit the operator framework. The result is that portions of both approaches appear in the article, and sometimes these are in conflict, leading to the lack of clarity that you have perceived. I don't see a quick fix to this problem – certainly it will take more than just adding new material. I would be in favor of getting rid of the operator framework and going back to the original formulation (and since that was unsourced material anyway, I see no problem with doing so.)
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several mathematical areas with slightly different meanings, but the same general "flavor". This article started out to be an umbrella article to cover all these specifics with fairly general language to emphasize the commonality. This meant that it needed to be described set-theoretically with respect to some generalized property
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terminology, and in order to fit the new framework, some classical terms had to be mangled. I am not convinced that this was a good thing to do. To get back to your specific point. In some cases there is a universal set that arises naturally. For instance, in topology, the whole topological space is
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Are you referring to the rewrite I did a few days ago? What about it do you not like? My main goal was to make it so that this page describes both the property called closure (a set satisfies this property if it is closed), as well as the closure operator (which maps each set to a closed set). If
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itself). However, in other situations, there may not be a clearly identified universal set. For instance, in dealing with the algebraic closure of a field, this, generally larger, unique field is constructed from the original field. In the original formulation (pre-operator), a closure operator was
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A discussion of how this relates (or doesn't) to closures in functional programming languages might be useful. I don't consider myself qualified to discuss this yet. Although the concept of mathematical closure has been brought up in some lectures on Lisp I've seen, the closures in the language
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My edits have, so far, been based solely on the definition of the first sentence of the article. However, you have raised a point that does need to be addressed, so I've now looked at the article as a whole, including its history, in order to respond. "Closure" is a concept that has been used in
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309:-Side comment from random person: while 3 - 8 is indeed not a natural number, it may be a bit confusing because natural numbers are often confused with integers. I would recommend changing this to 3 / 8, as this would make it clear that it's not a natural number nor integer.
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If a set were to be {0, 1}, would that be closed under addition? 0+1 = 0, ok, 0+0=0, ok, but 1+1=2. Does this mean that it is not closed under addition? All the examples I can find use natural numbers or integers as sets. These don't matter for addition or division.
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which would vary from one application to another. Then, in 2006, in an attempt to widen the applicability of the concept, the language about closure with respect to an operator on a set was introduced. Essentially this operator replaced the property
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Sure - Have at it. My interest is incidental anyway and I was just thinking to clarify the section. But I can see how that was a bit limiting. Better someone who has a broader view do it, especially toward making the whole article consistent.
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is a third concept that is presented in the article, and can be viewed as a generalization a closure operation. The problem is that, very often, the closure operator is much more difficult to specify than the closure operation. For example, the
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Somewhat late answer: “closure” in programming means something else (“having no open bindings” rather than “containing all results of an operation on its members”, which can only apply to a set), but I don’t think it is worth mentioning —
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I was confused by the previous version of the article. I have cleaned the article up, and it appeards from the resulting version that there is no need to split the article. On the opposite, one could discuss a merge here of
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set that contains all the subsets in question, including the closures. The definitions of closure and closure operator surely require one? Given the first sentence of this section, I was at first under the impression that
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520:("addition mod 2"). In this case, the answer is "yes". For closure properties of sets of integers under addition more generally, the entries on modular arithmetic and cyclic groups seem like reasonable places to start.
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Closure property is a property which a set either has or lacks with respect to a given operation.A set is closednwith respect to the operation if the operation can always be completed with elements in the set .
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As you point out, 1+1 = 2 ∉ {0,1}, so, no, our set is not closed under the usual addition operation for integers. In fact, no finite, nonempty set of integers other than {0} is closed under integer addition: let
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seem like a more concrete, perhaps separate concept from mathematical closure... but I could be wrong. In other words, is "closure" in functional programming overloaded to mean something else, or not?
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Hi, I speak spanish, so forgive my english. This article is not linked with spanish version. the spanish version is "Ley de composición interna". I don't know how to link the pages. Thanks
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Moreover, even if the results that I have sketched above are implied by the content of existing articles, these articles are much too technical for a non-expert who needs them.
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You're right about that. I'm glad you agree with the article, which says in the first sentence that "For example, the natural numbers are not ". Thank you for your help. -
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should be avoided, and some source should be cited, and some application example from term rewriting (where congruence closure is an important notion) should be given.
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results when the operation is applied on different occasions - as when there is an internal state which affects the result of a function. Looks to me like the word
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enough of a mathematician to know, please either clarify what unique might mean in this context & remove this comment, or eliminate the offending word.
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For example, a subset of a group is a subgroup if it is closed under the group operation (arity 2), the inverse operation (arity 1) and identity (arity 0).
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member, regardless of the arguments. The only other meaning I can ascribe is that this is in distinction to operations on sets which produce
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generated by a set of points is trivially defined as a closure operation, while the corresponding closure operator requires the concept of
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Many variations are possibles, but we lack clearly of elementary treatment of these subjects. Also, a large part, if not all, of
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Natural numbers are not closed in subraction because one (natural number) minus (natural number) is zero (not a natural number).
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I suggest to add the usual lemmas about closures of binary relations in this section. A first draft version is the following.
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The improvements appear to have been deleted, now. The uncorrect statement about closure in topology is still here.--
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This artice is very confusing because it tries to present two different concepts as if they were the same concept.
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was such a set. But now the last sentence suggests it isn't -- so what is? Moreover, the closure of a subset of
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is neccessary to apply the notions from the introductory sectiion to binary relations. Therefor, I replace "
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I agree with your points and have attempted to address them in the article. How do you like it now? -
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Also, I find that this article is written in a way which is somehow unnecessarily complicated... —
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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I'm not a mathematician, but surely this is not right. Surely the operation could only produce a
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that contains these points, while the corresponding closure operator can hardly be defined.
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Just some comments: For a WP article about a main result w.r.t. the first concept, try
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1\land x_{1}{\mathrel {R}}x_{2}{\mathrel {R}}\ldots {\mathrel {R}}x_{n}\}}" /: -->
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I have left 3 - 8, but added text to clarify for beginners why -5 is not natural.
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Modulo systems can contain no number equal to or greater than their modulus, so
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1\land x_{1}{\mathrel {R}}x_{2}{\mathrel {R}}\ldots {\mathrel {R}}x_{n}\}}": -->
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It'd be nice to see some mention made of this (even if only in passing)...
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The notion of a closure can be generalized for an arbitrary binary relation
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1\land x_{1}{\mathrel {R}}x_{2}{\mathrel {R}}\ldots {\mathrel {R}}x_{n}\}}
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I think part of the problem we are having here is there is no apparent
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a set is closed if and only if it is closed under taking the limits of
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performance of that operation on members of the set always produces a
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He’s right that they are not closed, but not that 0 is not a natural.
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I know, this draft certainly needs to be improved. The notion of
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you have some specific complaints, I could try to address them. -
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Indeed. I changed the text accordingly. Thanks for noticing. -
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Some important particular closures can be obtained as follows:
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On the other hand, define an operation + on our set {0,1} by
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I think that the notion of "closure without qualifier", i.e.
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And now it is not there any more: I see it was a link to
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Any of these four closures preserves symmetry, i.e., if
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An operation of a different sort is that of taking the
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1552:
1550:
1549:
1524:
1522:
1521:
1516:
1505:
1504:
1479:
1477:
1476:
1471:
1456:
1454:
1453:
1448:
1437:
1436:
1405:
1403:
1402:
1397:
1385:
1383:
1382:
1377:
1366:
1365:
1334:
1332:
1331:
1326:
1324:
1323:
1298:
1296:
1295:
1290:
1274:
1272:
1271:
1266:
1254:
1252:
1251:
1246:
1234:
1232:
1231:
1226:
1206:
1205:
1115:
1114:
1081:
1078:
1077:
1071:
1066:
1065:
1056:
1055:
1046:
1045:
1039:
1038:
1029:
1028:
1022:
1021:
994:
993:
981:
980:
947:
946:
918:
916:
915:
910:
902:
901:
853:
852:
825:
823:
822:
817:
805:
803:
802:
797:
744:
743:
666:
587:
560:
427:
382:closure operator
327:
263:
185:
134:
133:
130:
127:
124:
103:
98:
97:
87:
80:
79:
74:
66:
59:
42:
33:
32:
25:
24:
16:
2868:
2867:
2863:
2862:
2861:
2859:
2858:
2857:
2808:
2807:
2776:
2660:Zariski closure
2652:affine subspace
2631:
2627:
2623:
2619:
2615:
2611:
2607:
2603:
2599:
2591:
2587:
2583:
2579:
2575:
2552:First concept:
2547:
2504:
2405:
2350:
2349:
2324:
2323:
2268:
2246:
2224:
2196:
2188:
2187:
2142:
2120:
2092:
2070:
2062:
2061:
2021:
2020:
1950:
1949:
1900:
1899:
1880:
1879:
1834:
1812:
1790:
1768:
1760:
1759:
1740:
1739:
1697:
1675:
1653:
1645:
1644:
1625:
1624:
1605:
1604:
1571:
1563:
1562:
1535:
1527:
1526:
1490:
1482:
1481:
1462:
1461:
1422:
1408:
1407:
1388:
1387:
1351:
1337:
1336:
1309:
1301:
1300:
1281:
1280:
1257:
1256:
1237:
1236:
1094:
1086:
1085:
1057:
1030:
1013:
985:
972:
932:
923:
922:
838:
830:
829:
808:
807:
729:
721:
720:
672:
649:
614:
570:
566:
543:
539:
423:
346:
310:
264:—The preceding
260:
225:, referring to
219:
183:
171:
131:
128:
125:
122:
121:
99:
92:
72:
43:on Knowledge's
40:
30:
12:
11:
5:
2866:
2864:
2856:
2855:
2850:
2845:
2840:
2835:
2830:
2825:
2820:
2810:
2809:
2806:
2805:
2775:
2772:
2752:
2751:
2710:
2709:
2702:
2695:
2685:
2640:generating set
2630:and satisfies
2626:that contains
2546:
2543:
2542:
2541:
2503:
2500:
2484:
2483:
2404:
2401:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2337:
2334:
2331:
2301:
2298:
2295:
2292:
2289:
2286:
2281:
2278:
2275:
2271:
2267:
2264:
2259:
2256:
2253:
2249:
2245:
2242:
2237:
2234:
2231:
2227:
2223:
2220:
2215:
2212:
2209:
2206:
2203:
2199:
2195:
2175:
2172:
2169:
2166:
2163:
2160:
2155:
2152:
2149:
2145:
2141:
2138:
2133:
2130:
2127:
2123:
2119:
2116:
2111:
2108:
2105:
2102:
2099:
2095:
2091:
2088:
2083:
2080:
2077:
2073:
2069:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1887:
1867:
1864:
1861:
1858:
1855:
1852:
1847:
1844:
1841:
1837:
1833:
1830:
1825:
1822:
1819:
1815:
1811:
1808:
1803:
1800:
1797:
1793:
1789:
1786:
1781:
1778:
1775:
1771:
1767:
1747:
1727:
1724:
1721:
1718:
1715:
1710:
1707:
1704:
1700:
1696:
1693:
1688:
1685:
1682:
1678:
1674:
1671:
1666:
1663:
1660:
1656:
1652:
1632:
1612:
1603:for arbitrary
1590:
1587:
1584:
1581:
1578:
1574:
1570:
1548:
1545:
1542:
1538:
1534:
1514:
1511:
1508:
1503:
1500:
1497:
1493:
1489:
1469:
1446:
1443:
1440:
1435:
1432:
1429:
1425:
1421:
1418:
1415:
1395:
1386:; for example
1375:
1372:
1369:
1364:
1361:
1358:
1354:
1350:
1347:
1344:
1322:
1319:
1316:
1312:
1308:
1288:
1277:
1276:
1264:
1244:
1224:
1221:
1218:
1215:
1212:
1209:
1204:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1118:
1113:
1110:
1107:
1104:
1101:
1097:
1093:
1083:
1069:
1064:
1060:
1054:
1049:
1044:
1037:
1033:
1027:
1020:
1016:
1012:
1009:
1006:
1003:
1000:
997:
992:
988:
984:
979:
975:
971:
968:
965:
962:
959:
956:
953:
950:
945:
942:
939:
935:
931:
920:
908:
905:
900:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
865:
862:
859:
856:
851:
848:
845:
841:
837:
827:
815:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
750:
747:
742:
739:
736:
732:
728:
711:
710:
707:
692:
681:generalization
671:
668:
613:
610:
609:
608:
565:
562:
550:69.111.182.182
538:
535:
518:
517:
514:
511:
508:
422:
419:
418:
417:
416:
415:
366:
365:
345:
342:
341:
340:
317:99.231.244.178
307:
306:
292:
259:
256:
255:
254:
218:
215:
214:
213:
195:
170:
167:
164:
163:
160:
159:
156:
155:
144:
138:
137:
135:
118:the discussion
105:
104:
88:
76:
75:
67:
55:
54:
48:
26:
13:
10:
9:
6:
4:
3:
2:
2865:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2834:
2831:
2829:
2826:
2824:
2821:
2819:
2816:
2815:
2813:
2804:
2800:
2796:
2792:
2791:
2790:
2789:
2785:
2781:
2773:
2771:
2770:
2766:
2762:
2758:
2750:
2746:
2742:
2738:
2734:
2733:
2732:
2731:
2727:
2723:
2719:
2715:
2707:
2703:
2700:
2696:
2694:
2690:
2686:
2683:
2679:
2675:
2674:
2673:
2670:
2667:
2665:
2664:algebraic set
2661:
2657:
2653:
2648:
2643:
2641:
2637:
2610:that contain
2597:
2573:
2568:
2566:
2564:
2558:
2557:
2550:
2544:
2540:
2536:
2532:
2528:
2524:
2523:
2522:
2521:
2517:
2513:
2509:
2501:
2499:
2498:
2494:
2490:
2482:
2478:
2474:
2469:
2465:
2461:
2457:
2452:
2447:
2442:
2441:
2440:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2410:
2402:
2400:
2399:
2395:
2391:
2387:
2373:
2370:
2364:
2361:
2358:
2335:
2332:
2329:
2321:
2316:
2313:
2287:
2279:
2276:
2273:
2269:
2265:
2257:
2254:
2251:
2247:
2243:
2235:
2232:
2229:
2225:
2221:
2210:
2207:
2204:
2201:
2197:
2193:
2186:, but not in
2161:
2153:
2150:
2147:
2143:
2139:
2131:
2128:
2125:
2121:
2117:
2106:
2103:
2100:
2097:
2093:
2089:
2081:
2078:
2075:
2071:
2067:
2044:
2041:
2035:
2029:
2000:
1997:
1991:
1985:
1979:
1973:
1970:
1967:
1958:
1955:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1908:
1885:
1853:
1845:
1842:
1839:
1835:
1831:
1823:
1820:
1817:
1813:
1809:
1801:
1798:
1795:
1791:
1787:
1779:
1776:
1773:
1769:
1765:
1716:
1708:
1705:
1702:
1698:
1694:
1686:
1683:
1680:
1676:
1672:
1664:
1661:
1658:
1654:
1650:
1630:
1585:
1582:
1579:
1576:
1572:
1568:
1546:
1543:
1540:
1536:
1532:
1509:
1501:
1498:
1495:
1491:
1487:
1467:
1458:
1441:
1433:
1430:
1427:
1423:
1419:
1416:
1413:
1393:
1370:
1362:
1359:
1356:
1352:
1348:
1345:
1342:
1320:
1317:
1314:
1310:
1306:
1286:
1262:
1216:
1213:
1210:
1207:
1202:
1197:
1194:
1185:
1182:
1179:
1176:
1173:
1167:
1164:
1158:
1155:
1152:
1149:
1146:
1140:
1131:
1128:
1125:
1119:
1108:
1105:
1102:
1099:
1095:
1091:
1084:
1062:
1058:
1052:
1047:
1042:
1035:
1031:
1025:
1018:
1014:
1010:
1007:
1004:
1001:
998:
990:
986:
982:
977:
973:
963:
960:
957:
951:
943:
940:
937:
933:
929:
921:
903:
898:
893:
890:
884:
881:
878:
869:
866:
863:
857:
849:
846:
843:
839:
835:
828:
813:
790:
787:
784:
781:
775:
772:
769:
760:
757:
754:
748:
740:
737:
734:
730:
726:
719:
718:
717:
714:
708:
705:
701:
697:
693:
690:
686:
682:
678:
674:
673:
669:
667:
665:
661:
657:
653:
647:
642:
640:
636:
632:
628:
623:
622:
620:
607:
603:
599:
595:
590:
589:
588:
586:
582:
578:
574:
563:
561:
559:
555:
551:
547:
536:
534:
533:
529:
525:
524:75.184.118.88
521:
515:
512:
509:
506:
505:
504:
501:
499:
495:
491:
487:
483:
479:
475:
471:
467:
463:
459:
455:
449:
447:
443:
439:
435:
431:
420:
414:
410:
406:
402:
401:
400:
397:
393:
392:
391:
390:
387:
383:
379:
375:
371:
363:
359:
355:
351:
350:
349:
343:
338:
334:
330:
329:
328:
326:
322:
318:
314:
305:
301:
297:
293:
291:
288:
284:
283:
282:
279:
275:
271:
267:
257:
253:
250:
245:
244:
243:
242:
239:
235:
230:
228:
224:
216:
212:
208:
204:
200:
196:
194:
190:
186:
180:
176:
175:
174:
168:
153:
149:
148:High-priority
143:
140:
139:
136:
119:
115:
111:
110:
102:
96:
91:
89:
86:
82:
81:
77:
73:High‑priority
71:
68:
65:
61:
56:
52:
46:
38:
37:
27:
23:
18:
17:
2777:
2753:
2711:
2671:
2668:
2658:. Also, the
2644:
2614:and satisfy
2598:of a subset
2595:
2571:
2569:
2560:
2553:
2551:
2548:
2507:
2505:
2485:
2467:
2463:
2459:
2455:
2450:
2445:
2425:
2421:
2417:
2413:
2408:
2406:
2388:
2319:
2317:
2314:
1459:
1279:We say that
1278:
715:
712:
688:
684:
680:
676:
650:— Preceding
645:
643:
638:
634:
630:
626:
624:
618:
616:
615:
612:Why 'Unique'
577:98.207.0.180
567:
540:
522:
519:
502:
497:
493:
489:
485:
481:
477:
473:
469:
465:
461:
457:
453:
450:
424:
405:78.15.196.42
367:
347:
311:— Preceding
308:
261:
231:
220:
178:
172:
147:
107:
51:WikiProjects
34:
2489:Daren Cline
2430:Daren Cline
685:generalized
571:—Preceding
544:—Preceding
500:is finite.
428:—Preceding
344:Closed sets
227:closed sets
123:Mathematics
114:mathematics
70:Mathematics
2812:Categories
2019:, we have
694:The link "
472:+ . . . +
358:closed set
217:I think...
2676:Redirect
2409:universal
656:82.0.88.8
635:different
516:1 + 1 = 0
513:1 + 0 = 1
510:0 + 1 = 1
507:0 + 0 = 0
39:is rated
2761:D.Lazard
2722:D.Lazard
2586:satisfy
652:unsigned
598:PJTraill
573:unsigned
546:unsigned
484:, since
442:contribs
430:unsigned
362:topology
333:PJTraill
313:unsigned
296:PJTraill
278:contribs
266:unsigned
203:PJTraill
2780:Ilya239
2697:Create
2687:Create
2596:closure
2563:variety
702:" and "
689:applied
644:If you
627:unique
464:. Then
434:Nswartz
378:filters
150:on the
41:C-class
2574:: Let
687:" by "
639:unique
619:unique
386:Kompik
184:occono
179:Added.
47:scale.
2704:Keep
2527:WP:RS
1335:, if
1005:: -->
396:lethe
287:lethe
270:Ieopo
249:lethe
28:This
2799:talk
2784:talk
2765:talk
2745:talk
2726:talk
2535:talk
2516:talk
2493:talk
2477:talk
2434:talk
2394:talk
2348:and
2315:---
1948:and
713:---
660:talk
631:same
602:talk
581:talk
554:talk
528:talk
496:and
438:talk
409:talk
374:nets
337:talk
321:talk
300:talk
274:talk
238:Talk
207:talk
189:talk
181:----
142:High
2680:to
2602:of
2531:JBL
646:are
376:or
280:) .
234:MFH
2814::
2801:)
2786:)
2767:)
2759:.
2747:)
2728:)
2720:.
2645:A
2642:.
2537:)
2518:)
2495:)
2479:)
2436:)
2396:)
2371:∈
2368:⟩
2356:⟨
2312:.
2214:Σ
2110:Σ
2048:⟩
2027:⟨
2004:⟩
1983:⟨
1977:⟩
1965:⟨
1906:Σ
1746:Σ
1611:Σ
1589:Σ
1457:.
1243:Σ
1220:Σ
1217:∈
1211:∧
1195:∣
1192:⟩
1186:…
1174:…
1159:…
1147:…
1138:⟨
1132:∪
1112:Σ
1048:…
1011:∧
999:∣
996:⟩
970:⟨
964:∪
891:∣
888:⟩
876:⟨
870:∪
788:∈
782:∣
779:⟩
767:⟨
761:∪
706:".
691:".
662:)
604:)
583:)
556:)
530:)
492:+
488:≠
478:ne
476:=
468:+
444:)
440:•
411:)
323:)
302:)
276:•
209:)
191:)
2797:(
2782:(
2763:(
2743:(
2724:(
2632:P
2628:X
2624:S
2620:X
2616:P
2612:X
2608:S
2604:S
2600:X
2592:P
2588:P
2584:S
2580:S
2576:P
2533:(
2514:(
2491:(
2475:(
2468:X
2464:X
2460:X
2456:X
2451:P
2446:P
2432:(
2426:X
2422:X
2418:X
2414:X
2392:(
2374:R
2365:y
2362:,
2359:x
2336:y
2333:R
2330:x
2300:)
2297:)
2294:)
2291:)
2288:R
2285:(
2280:f
2277:e
2274:r
2270:l
2266:c
2263:(
2258:m
2255:y
2252:s
2248:l
2244:c
2241:(
2236:n
2233:r
2230:t
2226:l
2222:c
2219:(
2211:,
2208:b
2205:m
2202:e
2198:l
2194:c
2174:)
2171:)
2168:)
2165:)
2162:R
2159:(
2154:f
2151:e
2148:r
2144:l
2140:c
2137:(
2132:m
2129:y
2126:s
2122:l
2118:c
2115:(
2107:,
2104:b
2101:m
2098:e
2094:l
2090:c
2087:(
2082:n
2079:r
2076:t
2072:l
2068:c
2045:c
2042:,
2039:)
2036:a
2033:(
2030:f
2007:}
2001:c
1998:,
1995:)
1992:b
1989:(
1986:f
1980:,
1974:b
1971:,
1968:a
1962:{
1959:=
1956:R
1936:}
1933:f
1930:,
1927:c
1924:,
1921:b
1918:,
1915:a
1912:{
1909:=
1886:S
1866:)
1863:)
1860:)
1857:)
1854:R
1851:(
1846:f
1843:e
1840:r
1836:l
1832:c
1829:(
1824:m
1821:y
1818:s
1814:l
1810:c
1807:(
1802:b
1799:m
1796:e
1792:l
1788:c
1785:(
1780:n
1777:r
1774:t
1770:l
1766:c
1726:)
1723:)
1720:)
1717:R
1714:(
1709:f
1706:e
1703:r
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1695:c
1692:(
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1670:(
1665:n
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1659:t
1655:l
1651:c
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1586:,
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1547:n
1544:r
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1513:)
1510:R
1507:(
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1496:x
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1445:)
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1439:(
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1428:s
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1420:c
1417:=
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1394:R
1374:)
1371:R
1368:(
1363:x
1360:x
1357:x
1353:l
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1346:=
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1287:R
1275:.
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1135:{
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1100:e
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1008:1
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991:n
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958:=
955:)
952:R
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894:x
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879:y
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272:(
236::
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154:.
53::
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