A B (A ´ B) , , . S - K . S ´ K (s, k), : s k. , .
.
1. ( ) r A B A ´ B, ..
.
, A=B ( rÍ A 2), , r A.
a b ( ) r.
. , : r, t, j, s, w ..
2. r D r={ a | $ b, a r b } ( ). r R r={ b | $ a, a r b } ( ).
1. A ={1; 3; 5; 7} B ={2; 4; 6}. t={(x; y)Î A ´ B | x+y =9}. (3; 6), (5; 4) (7; 2), t={(3; 6), (5; 4), (7;2)}. D t={3; 5; 7} R t= B ={2; 4; 6}.
,
2. r={(x; y) | x y x y }. =. , D r= R r.
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3. A , B . j={(x; y)Î A ´ B | y x } A B.
,
3.1., , t={(x; y)Î A ´ B | x+y =9}, t={(3; 6), (5;4), (7;2)}. , ( ) . .
:
1) ;
2) ;
3) : A B r . . r , , . , , , .
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4) : A ={ a 1, a 2, , an } B ={ b 1, b 2, , bm }, r A ´ B. r M =[ mij ] n ´ m,
.
, .
4. A ={1; 3; 5; 7} B ={2; 4; 6}. t={(x; y) | x+y =9}. , , .
. 1) t={(3; 6), (5; 4), (7; 2)} - ;
2) .
3)
.,
.
3. n- (n - ) r n , ()
rÍ A 1´´ An ={(a 1, , an)| a 1Î A 1Ù Ù an Î An }
. n - r. . , . , , .
relation, . R () r ().
() , .
4. rÍ A ´ B A ´ B. r-1 r A ´ B, :
r-1={(b, a) | (a, b)Îr}.
5. r Í A ´ B A ´ B, s Í B ´ C B ´ C. sr t Í A ´ C, :
t=s◦r= {(a, c)| $ b Î B, (a, b)Îr (b, c)Îs}.
5. , C ={,,!, d, à}. r A ´ B s B ´ C :
r={(1, x), (1, y), (3, x)};
s={(x,,), (x,!), (y, d), (y, à)}.
r-1 s◦r, r◦s.
. 1) r-1={(x, 1), (y, 1), (x, 3)};
2) ,
s◦r={(1,,), (1,!), (1, d), (1, à), (3,,), (3,!)},
(1, x)Îr (x,,)Îs (1,,)Îs◦r;
(1, x)Îr (x,!)Îs (1,!)Îs◦r;
(1, y)Îr (y, d)Îs (1, d)Îs◦r;
(3, x)Îr (x,!)Îs (3,!)Îs◦r.
3) r◦s=Æ.
,
1. :
1) ;
2) ;
3) - .
. 1 .
2. , , , . (a; b) Î (s◦r)-1 Û (b; a) Î s◦r Û $ c , (b; c) Î r (c; a) Î s Û $ c , (c; b) Î r-1 (a; c) Î s-1 Û (a; b) Î r -1◦s -1.
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