R Ω R Ω ´ Ω ( . 3-2). á x, y ñ R, .. á x, y ñÎ R, xRy, : x R y. xRy x y, x y ..
( Æ), Ω, .. . - ( U), Ω. ( ), Ω, : xy ↔ x = y ( ↔ ). ( ), Ω, : x y ↔ x ¹ y.
Ω Ω ´ Ω, - - : R 1 R 2 (), R 1 R 2 (), ( U). - (. 3-4), , . . .
R R -1, : xR -1 y ↔ yRx. R 1 R 2 , R 1 R 2, - : x (R 1 R 2) y, z Î Ω, xR 1 z zR 2 y. R 1 , R 2 , R 1 R 2 - , .. . , -, (A B) C = A (B C), A B C .; Rn.
. Rd = = ()-1 R.
1. R Ω = { a, b, c, d }: aRb, bRc, cRd, dRa, aRa, cRc, bRd, dRb. Rd, R.
1. , R. - , R. Ω ( ) , - R ( ). Ω = { a, b, c, d }, :
Ω ´ Ω ={á a, a ñ,á a, b ñ,á a, c ñ,á a, d ñ,á b, a ñ,á b, b ñ,á b, c ñ,á b, d ñ,á c, a ñ,á c, b ñ,á c, c ñ,á c, d ñ,á d, a ñ,á d, b ñ,á d, c ñ,á d, d ñ}
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( 16 ).
R :
R = {á a, b ñ,á b, c ñ,á c, d ñ,á d, a ñ,á a, a ñ,á c, c ñ,á b, d ñ,á d, b ñ}
( 8 ). R , Ω ´ Ω R. , , , R. á a, a ñ. R ( 5- ) . á a, b ñ R ( 1- ) . á a, c ñ R . á a, d ñ R . ,
= {á a, c ñ,á a, d ñ,á b, a ñ,á b, b ñ,á c, a ñ,á c, b ñ,á d, c ñ,á d, d ñ}.
2. ()-1, . , , , (- á x, y ñá y, x ñ; á x, x ñ , ). , 1. , :
()-1={á c, a ñ,á d, a ñ,á a, b ñ,á b, b ñ,á a, c ñ,á b, c ñ,á c, d ñ,á d, d ñ}.
3. ()-1, 2, Rd ■
1. , .
01. Ω = { a, b, c, d, e, f }; aRa, aRd, aRe, aRf, bRa, cRc, bRc, fRc, bRe, eRa, eRc, eRf, fRa, dRf.
02. Ω = { a, b, c, d }; bRa, cRb, dRc, dRa, bRb, cRa, bRd, dRd.
03. Ω = { a, b, c, d, e, f }; aRa, aRd, aRe, aRf, bRa, cRc, bRc, fRc, bRe, eRa, eRc, eRf, fRa, dRf.
04. Ω = { a, b, c, d }; bRa, cRb, dRc, dRa, bRb, cRa, bRd, dRd.
05. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa.
06. Ω = { a, b, c, d, e }; aRb, bRc, aRd, bRa, eRd, cRd, dRc.
07. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa.
08. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRa.
09. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa.
10. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRa, eRe, eRd, cRd, dRc.
12. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRa, eRe, eRd, cRd, dRc.
13. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRa.
14. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa ■
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2.1. . -. , Ω, , , , Ω. xRy R , , x y y x.
R , x Î Ω xRx, -, x Î Ω xRx (.. x x). , : E Í R R Í .
, R Í R -1. , xRy , yRx. , R Ç R -1 = Æ. , - xRy yRx . -, . , ϸ Ը, Ը -; , , . -, R R -1Í . , xRy yRx -, x = y, : xRy yRx , x = y.
, R 2Í R, : xRy yRz -, xRz. R , . R , - .
, k = 1, 2,... Rk R -1 = Æ, -: x 1 Rx 2, x 2 Rx 3, , xk 1 Rxk , xk x 1.
, . -, - , , , z, - , , z (), z - x ().
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1. R , .
, x xRx, xR -1 x, , á x, x ñÎ R∩R -1, .. R∩R -1 ≠ Æ. , R . , xRx - 1 , , R ■
2. R Ω = { a, b, c, d, e } : aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRa. :
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a;
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1. aRa, .
2. bRb, . 1 - R ■
3. R Ω = { a, b, c, d, e } : aRb, bRb, aRc, bRa, eRe, eRd, cRd, dRc. :
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1. bRb, R .
2. R , 1 - R .
3. aRc cRd, R aRd. aRd , , ■
2.
01. R Ω = { a, b, c, d, e } : aRb, bRb, aRc, bRa, eRe, eRd, cRd, dRc. :
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02. R Ω = { a, b, c, d, e } : aRb, bRb, aRe, bRa, eRe, eRd, cRd, dRc. :
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03. R Ω = { a, b, c, d, e } : aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRa. :
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04. R Ω = { a, b, c, d, e } : aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRa. :
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a;
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05. R Ω = { a, b, c, d, e } : aRb, bRc, aRd, bRa, eRd, cRd, dRc. :
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06. R Ω = { a, b, c, d, e, f }: aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa. :
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07. R Ω = { a, b, c, d, e, f }: aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa. :
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08. R Ω = { a, b, c, d, e, f }: aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa. :
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09. R Ω = { a, b, c, d, e, f }: aRa, aRd, aRe, aRf, bRa, cRc, bRc, fRc, bRe, eRa, eRc, eRf, fRa, dRf. :
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10. R Ω = { a, b, c, d, e, f }: aRa, aRd, aRe, aRf, bRa, cRc, bRc, fRc, bRe, eRa, eRc, eRf, fRa, dRf. :
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11. R Ω = { a, b, c, d } : bRa, cRb, dRc, dRa, bRb, cRa, bRd, dRd. :
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12. R Ω = { a, b, c, d } : bRa, cRb, dRc, dRa, bRb, cRa, bRd, dRd. :
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. x y Ω R, : x y y x. R - IR: xIRy ↔ x y R. , . , IR = . , IR R.
4. IR R 1. R á a, b ñ,á b, c ñ,á c, d ñ,á d, a ñ,á a, a ñ,á c, c ñ,á b, d ñ,á d, b ñ. R −1 : R −1= {á b, a ñ,á c, b ñ,á d, c ñ,á a, d ñ,á a, a ñ,á c, c ñ,á d, b ñ,á b, d ñ}. , R R −1= {á a, a ñ,á a, b ñ,á a, d ñ,á b, a ñ,á b, c ñ,á b, d ñ,á c, b ñ,á c, c ñ,á c, d ñ,á d, a ñ,á d, b ñ,á d, c ñ}. IR = = {á a, ñ,á b, b ñ,á c, a ñ,á d, d ñ}. , IR , , ( á a, ñ á c, a ñ, á b, b ñ, á d, d ñ). , IR -; ( á a, ñ á c, a ñ, á a, a ñ); , R R −1 ( - á a, b ñ á b, c ñ, á a, ñ ), ■
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3. IR 1 ■
2.2. . Ω . - - Ω : x 1, x 2, ,, xn. xi xj , xiRxj Ω ( i = j (xi,xj) - xi). (. 6). () ; , .
, , . , ( , ). , - -. R, G (R).
, . - , . , . - , . - R Rd - .
1. G (R) , G (Rd) ; , G (R) , G (Rd) .
2. G (R) , G (Rd) - ; , G (R) , G (Rd) .
5. R 1 Ω = { a, b, c, d }: aRb, bRc, cRd, dRa, aRa, cRc, bRd, dRb. a, b, c, d , B, C, D, . G (R) R .1. - G (Rd) 1 .2.
.1. G (R) R
.2. G (Rd) Rd ■
4. 1 - ■
G (V, A) - R = R (G); Ω, , V G (V, A); xRy ↔ (x, y) A. , R (G) , G; , ( -) G G (R (G)) = G, . - .
5. G .6-10, , R (G) xRy ■
2.3. . - , . R (), -, . : ; - 3 ; - .
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Ω: Ω = (i ¹ j)→(Ωi Ωj = Æ). Ω - R: xRy , Ωi, - .
2. Ω , ■
, Ω Ω .
, , - .
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R, , - , Ω, ≠ , xRy yRx.
, R :
1. Ω: Ω = , , - xRy , , x, , - y.
2. IR .
1, 2 .
3. R IR - ■
6-8 3 Ω. - . , R . Ωi : Ω 0 Ω, , Ω xRy. , Ω 1 Ω Ω 0 , Ω W0 xRy. , Ω. , 6-8. , IR - Ωi (i = 1,..., m).
R-
- , ( - ) Ω. , . . - , . , - , , , - , . , ó .
. x Ω Rx xR x R:
Rx = { y Ω | yRx }, (1a)
xR = { y Ω | xRy }. (1b)
x R, Î Ω xRy, R, Î Ω x (.. yRx). - x , xR = Ω, , Rx = Æ.- 1- , , 2- - - , . - R ΩR, - ΩR. , - , . , -, . , -, x R. , x ( ), , , . , x . , ( ). , Ω.
6. R Ω = { a, b, c, d }: aRb, bRc, cRd, dRa, aRa, cRc, bRd, dRb 1. ΩR ΩR.
1. x , , , xRy y. , a aR; b bRa; c cRa; d dRc. , , ΩR = Æ.
2. x , , , , x R, , R. , . , : ΩR =Æ. ■
7. R Ω = { a, b, c, d }: aRb, bRb, cRd, dRa, aRa, cR, bRd, dRb ( 6) ΩR = { c }. , R, .. . , ■
G (R) R , - ; , . , R Rd 1 (. .1 2) . , R (. 6), - Rd .
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4. Ω
ΩR = , ΩR = ■ (2)
4, . - R. ΩR, -, R ; - R-.
, , , . , y y ≠ x. (2), . , , ( ), - 16-3.2.
3.1. ΩR. R -- , - . , , . - , , - . , ( ) .
, xRy, 1 3 5 7. . .
1. ΩR.
1. D = Æ.
2. xRy.
2.1. , y D. , - .
2.2. y D .
3. ΩR = Ω D.
8. R Ω = { a, b, c, d }: aRb, bRb, cRd, dRa, aRa, cR, bRd, dRb, 7. 1.
1. D = Æ.
2. xRy. :
1- aRb; b D. D = D { b } = { b };
2- bRb; b D;
3- cRd; d D. D = D { d } ={ b, d };
4- dRa; a D. D = D { a } ={ a, b, d };
5- dRa; a D;
6- aRa; a D;
7- bRd; d D;
8- dRb; b D.
3. ΩR = Ω D = { a, b, c, d } { a, b, d } = { c }
5. 1 ΩR -.
01. Ω = { a, b, c, d, e, f }; aRa, aRd, aRc, aRf, bRa, cRc, bRc, fRc, eRa, eRc, eRf, fRa, dRf.
02. Ω = { a, b, c, d }; bRa, cRb, dRc, dRa, bRb, cRa, bRc, dRb.
03. Ω = { a, b, c, d, e, f }; aRa, aRd, aRe, aRf, bRa, cRc, bRc, fRc, bRe, eRa, eRc, eRf, fRa, dRf.
04. Ω = { a, b, c, d }; bRa, cRb, dRc, dRa, bRb, cRa, bRd, dRd.
05. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd..
06. Ω = { a, b, c, d, e }; aRb, bRc, aRd, bRa, eRd, cRd, dRc.
07. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRa.
08. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRc.
09. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRd, eRb, aRc, bRf, fRc, cRd, cRc, eRd.
10. Ω = { a, b, c, d, e }; aRb, bRb, aRc, eRe, eRd, cRd, dRb.
12. Ω = { a, b, c, d, e }; aRa, bRc, aRc, bRe, eRe, eRd, cRd, dRc.
13. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRe, eRe, eRd, cRd, dRa.
14. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRd, cRc, eRd, fRf.
15. Ω = { a, b, c, d, e }; aRb, bRb, bRa, eRe, eRd, cRd.
16. Ω = { a, b, c, d, e }; aRb, bRb, aRe, eRe, eRd, cRd, dRc.
17. Ω = { a, b, c, d, e }; aRc, bRe, eRe, eRd, cRd, dRa.
18. Ω = { a, b, c, d, e }; aRb, bRb, aRc, bRe, eRe, dRa.
19. Ω = { a, b, c, d, e }; aRb, bRc, aRd, bRa, eRd, cRd, dRc.
20. Ω = { a, b, c, d, e, f }; bRd, cRf, aRc, bRf, fRc, cRd, cRc, eRd, fRa.
21. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, bRf, fRb, cRd, eRd, fRa.
22. Ω = { a, b, c, d, e, f }; aRb, bRd, cRf, dRe, eRb, aRc, bRf, fRc, cRc, fRa.
23. Ω = { a, b, c, d, e, f }; aRd, aRe, aRf, cRc, bRc, fRc, bRe, eRc, eRf, fRf, dRf.
24. Ω = { a, b, c, d, e, f }; aRa, aRd, aRe, aRf, bRa, cRc, bRc, fRc, bRe, eRa, eRc, eRf, fRa, dRf.
25. Ω = { a, b, c, d }; bRa, cRb, dRc, dRa, bRb, cRa, dRc.
26. Ω = { a, b, c, d }; bRa, cRb, dRc, dRa, bRb, cRa, bRd, dRd ■
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R -