3.
3.1.. .
3.2.. . .
3.3. .
. , ( ) , .
. , ,
.. . 1, . 1.8.-1.10
:
- ? ( ) ?
- ={a,b,c} B={x,y,z}. ?
- {(1,2), (2,4), (3,6),}
- = {(b, ), (, ), (d, i), (f, ){g, u)}
- = {(v, ), (w, ), (, i), (, )(z, u)}.
) -1 -1
) -1 .
) -1 .
6. 6
3.1.
X Y (x,y) , x X, y Y.
X Y
3.1.
, ,
X Y ≠ Y X
3.1.
X={1,2,3}, Y={#,*}
X Y = {(1,#), (1,*), (2,#), (2,*), (3,#), (3,*)}
Y X = {(#,1), (#,2), (#,3), (*,1), (*,2), (*,3)}
, , .
m , n , mn . , , , , .
. . .
3.2(1)
R ( Relation ) A B. (, b) R, aRb, , b R, , b. , , , a φ b a b
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3.2 (1)
1) A B
2) = {1,2,3}, = {*, #}, A B = { (1, *), (2,*), (3,*), (1,#), (2,#), (3,#)},
R = {(l,*), (l,#), (3,#)} . 3R#, (3,#) R. A B , 26 = 64 A B. , 64 A B.
3.2. (2)
= , A A .
3.2 (2).
1). N:
- ≤ (7,9) (7,7), (9,7);
- , , (6,9), (4,2), (2,4), (4,4), (7, 9) (9, 7);
- (. . aRb b) (2, 4) (4, 4), (4, 2) (7, 9).
2). :
- (3, 4) (2, 1/21), (3, 4) (1, 6); ;
- , ;
- x (1, 1) (2, 2), 1 = 2, 1 = -2.
3). :
; ; ; .
3.2.(3).
R , (,) R. , R R. R , (, ) R
. , R
R.
3.2.(4)
R R-1 . R . R-1 : R-1= {(b,a): (a,b) R}
, (b, ) R-1 , (a, b) R ,
, b R-1a , aRb. R-1 R.
3.2. (4)
1) R = {(l,*), (l,#), (3,#)}, R-1 = {(*, 1), (#,1), (#,3)}.
2) R - {(,): }, R-1 = R
3.2.(5)
R R , S B C - S B C.
S R T A C, :
T = {(a,c): b B, (a,b) R (b, c) R}
T T = S R
3.2.(5)
1) A={1,2,3}, B={#,*}, C={w,x,y,z}
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R S B C :
R={(1,#), (1,*),(3,#)}
S={(#,w), (#,x), (*,y), (*,z)}
T A C = S R = {(1,w), (1,x), (1,y), (1,z), (3,w), (3,x)}
(1,#) R (#,w) S (1,w) S R
(1,#) R (#,x) S (1,x) S R
(1,*) R (*,y) S (1,y) S R
2) R S , : S = {(x, x+2): x } R = {(x, x4): x } S R = {(x, x4 +2): x }
R S = {(x, (x+2)4): x }.
3.2.(6)
R , (, ) R .
R , (, b) R ≠b.
R , b, , (, b) R , (b,a) R.
R , , b , (, b) R (b,c) R, , (, c) R.
R , b A, (, b) (b, ) R , = b.
3.2.(6)
= {1,2,3,4} R1
R1 = {(1,1), (2,2), (3,3), (4,4), (1,2), (1,4), (2,1), (4,1)}.
R1 , .. , (,) R1. , (a,b) R1 (b,) R1, , R1 , ,
.
3.3.(1)
R , , .
3.3.(1)
. R2 :
R2 = {(, b): - b = 5 }( a b 5).
, (8,3) R2, 8-3 = 5 = 51, (-11,4) R2,
(-11)-4 = -15 = 5 (-3).
R2 . (.. ), = 0 = 5 0 = 5 = 0, (, ) R2.
R2 . , (, b) R2. t, b = 5 t b = ( b) = (5t) = 5(t) (t). , (b, ) R2
R2. , a, b , (, b) R2
(b, c) R2.
(, b) R2, a b = 5k k,
(b,c) R2, b c = 5m m.
, (a b) + (b c) = 5k + 5m = (a c) = 5(k+m), (k+m) .
, R2
.
. , ( ) , ,
( r = 0), . .
3.3.(2)
, A , A .
, , A . , , , , , . .
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, , . . , , , b , , , b > .
, , b > , . . b , , < b, . . , b. , , , .
. , , 1. , , b 1 b 1, , 1. , b, b , , , 1.
3.3.(6)
, .
:
. , Rb , b, R , .
, R , : R. . ,
b , . , b, .. R Rb. R Rb. , . . xR, aRb Rb, . . b.
, b . , b, , . A .
, R ,
. R.
. , R :
(, b) R , b . R , , . R :
(, b) R , b , .
3.5.(6)
[a] = {x: (x, a) R2} = {x: x R2 a} = {x: x a =5k k} =
= {x: x = a+5k k}
( 5):
[0] = {, - 15, - 10, - 5, 0, 5, 10, 15, 20, }
[1] = {, - 14, - 9, - 4, 1, 6, 11, }
[2] = {, - 13, - 8, - 3, 2, 7, 12, }
[3] = {, - 12, - 7, - 2, 3, 8, 13, }
[4] = {, - 11, - 6, - 1, 4, 9, 14,}
, R2 5 : [0], [1], [2], [3], [4].
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3.3.(3)
, , . , , ≤, .
, . , , <, .