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??. ???(Set Theory)????????.????????16???,???????????????.?????????? 19?? 70??????????(G . Cantor) ??????????????????????.???????????????????????,?????????????????????,???????????.??,?????????????.?????????,?????????????????,?????,?????????????????,??????(B . A . W . Russell)??,?????????????????.. .

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    1. ???

    2. 1. ??????? ??(????) ???????? ?????????? ?????????????

    3. ?? ???(Set Theory)????????.????????16???,???????????????.?????????? 19?? 70??????????(G . Cantor) ??????????????????????.???????????????????????,?????????????????????,???????????.??,?????????????.?????????,?????????????????,?????,?????????????????,??????(B . A . W . Russell)??,?????????????????.

    4. ?????????????????????????????????,????????????ZFS(E . Zermelo, A . Fraenkel, T . Skolem)?NBG(Von Neurnann, P . Bernavs, K . Gödel)?????????. ? 20?? 60??,P . L . Cohen???????????????????????????,???????????,????.?????,????? L . A.Zadeh???Fuzzy???, ?? 20??80???????Z . Pawlak???Rough???,??????????????, ??????????,????????????,???????.???????????????????????.

    5. ???????????????????????????????????.???????????????,?????????????,????????????.

    6. ??(????) ???????????(?????????,?A,B?),????????? (?????????,?x,y?). x?A?????: x?A (??x??A) x??A?????: x?A (??x???A) ????????,???????A,????x, x?A?x?A?????????.

    7. ??10??????A={1, 3, 5, 7, 9} ?????????????????{a, 2, Fred, New Jerseg} ??B={x | x???10????} ?????A=B

    8. ???????? ????:??????A?B????????,????A?B??,??A= B?B=A. ???: ????B???????A???,B??A????(????). ??B?A (??B???A),?A?B (??A??B). ??: A= B????A?B?A?B.

    9. ?????????? ?A?????,???B??????????:B={x?A | P(x)}, ??P(x)??x?B???????? ??:?????????????,??????,??????????: ?={x?A | x ?x} ???????,????????????????,??,?????. ??????????. ???E??.

    10. ????? ??4 ?S?????S???n?????,n?????,??S?????,?n???S???,?| S |??? ? ?A???10??????,?| A | = 5 ??F????,??| F | = 0 ??5 ???????????,????????

    11. ??? ???????????????????????,???????????????????????????,?????????,??S????????????

    12. ??? ??6 ????S, S???????S???????,?P(S)??? ? ??{0, 1, 2}?????P({0,1,2}={F,{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}? ????????????{F}????????

    13. ??? ????????????{F}???????? P(F)={F} P({F})={F,{F}} ???????n???,????????2n????

    14. 2. ???? ??7 ??n??(a1,a2,…,an)??a1??????,a2??2???,…,an??n???????? 2????????? ??A?B?????C=A?B???????(a,b)???, ??a?A, b?B. ??? A?B={(a,b) | a?A? b?B}

    15. ???? ? ?A???????????,B????????????????A?B?????A×B???? ?:????A×B???(a, b)????????,??a???,?b???????????A×B????????????????????

    16. ???? ? A={1,2}, B={a, b, c}, A×B=? ???A×B????B×A

    17. ???? ??9 ??A1, A2,…,An??????A1×A2ׅ×An??,????n??(a1,a2,…,an)???,????i=1,2,…,n,ai?Ai? ???????A×B×C,??A={0,1},B={1,2},C={0,1,2}?

    18. ???? ???????A×B×C,??A={0,1},B={1,2},C={0,1,2}? ?:A×B×C={(0,1,0),(01,1),(0,1,2),(0,2,0),(0,2,1),(0,2,2),(1,1,0),(1,1,1),(1,1,2),(1,2,0),(1,2,1),(1,2,2)}

    19. 3. ???? ???? ???? ????????(?)?? ???????

    20. ???? ??A?B????A?B??????????, ??A?B, ??? A?B={x | x?A?x?B} ????????: ??? A ? B= B ? A ???A ? (B ? C)= (A ? B) ? C

    21. ???? ??A?B??????A?B??????????, ??A ? B, ??? A ? B={x | x?A?x?B} ????????: ??? A ? B= B ? A ???A ? (B ? C)= (A ? B) ? C

    22. ????????(?)?? ??????:???A?????B??????????????A???B??,??A-B, ??? A-B={x?A | x?B} ????U,??A?U, U-A??A??U???,?U??????,???A???,??

    23. ????

    24. ??????? ?????????????????????????????????????,??????????????????????????1???????????,?0????????????

    25. ??????? ? ?????? An(B?C)=(A nB) ? (AnC)

    26. ???????? ????? ??:????????????????????,??????????????

    27. ???????? ??????????????????????? ????U????,???U???????????,??a1,a2,…,an?????????n?????U???A:??ai??A,?????i??1;??ai???A,?????i??0?

    28. ???????? ? ?U={1,2,3,4,5,6,7,8,9,10},??U?????????,?ai=i???U????????????????????5??????????????? ?:U?????????{1,3,5,7,9}???,?1,3,5,7,9??1,????0,? 10101 01010

    29. ???????? ? ??{1,2,3,4,5}?{1,3,5,7,9}??????11111 00000?10101 01010,?????????????? ?:????????????? 11111 00000 ? 10101 01010 = 11111 01010,????{1,2,3,4,5,7,9} ???????????? 11111 00000 ? 10101 01010= 10101 00000,????{1,3,5}

    30. 6. ???? ????? ??????? ??? ?? ????

    31. ????? ???????: ?????? ????????--??: ??????A?B??????,??A?B????????.??A~B. ?????: 1. ???: A~A; 2. ???: A~B?B~A; 3. ???: A~B, B~C?A~C.

    32. ??????? ???: 0:=?, 1:={?} ????: n+1:={0,1,…,n} ????: N={0,1,…}, N+ ={1,2,…} ???: ????A????????,??A????. ??: ?????? ????????????. ???????????????

    33. ???(I) ???: ????????????? ???: ?????N?????????? ??1. ????Q={m/n | m?Z, n?N+}???? ??:????h=|m|+n ??2. ??????????????

    34. ???(II) ??3. ????????????????????? ??: ??????. ??1. ??????????? ??2.?A????, B???,?A?B~B ??: ?????????????????????????(at most countable) ????(enumeratable),???????

    35. ???? ????: ????N?????????? ????: ?P(N)?????????????? ??3. F={f: N?{0,1}}?P(N)?? ??4. X={A?P(N) | ?k?N, ?n?k, n?A}???? ??5. ?????1=0.11….(1??)

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