. . . .
: [1] . 67-89; [2] . 87-104; [3] . 38-57; [4] . 28-42; [5] . 95-103; [6] . 76-82; [7] . 22-36.
( I )
1. ={5, 7, 8, 9, 10} ={3, 4, 5, 6, 9}. R: 2 , Î, Î. R: ) ={(3, 5), (5, 7), (6, 8), (9, 10)}; ) ={(5, 3), (7, 5), (8, 6)}; ) R: =2+, Î, Î; ) ={(5, 3), (7, 4), (7, 5), (7, 6), (7, 3), (8, 4), (8, 3), (8, 5), (8, 6)}.
1. ={2, 4, 6, 8, 10, 12}, R: 4 , , Î. R:
)2 4 ) 2 4 ) 2 4 )2
12 6 12 4
12 6 6 6
10 8 12
10 8 10 8
10 8
2. , , Î, Î. , (, )Î: ) Ì; ) Ì; ) Ì´; ) (´)Ç=; ) (´)Ç= ´; ) (´)È=; ) (´)\=Æ
2. ={|ÎZ, 13££2} R: =2. :
) (-6, -3)ÎR; ) (-3, -6)ÎR; ) (-4, -2)ÎR;
) (-8, -4)ÏR; ) -12R-6; ) 2R1.
3. ={, , } ={, , , , }. R: , Î, Î. R:
) ={(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )};
) R
) ={(, ), (, ), (, ), (, ), (, ), (, ), (, )};
) ={(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )}.
3. ={ | ÎN, ≤6} :{(6, 6), (6, 3), (6, 2), (6, 1), (5, 5), (5, 1), (4, 4), (4, 2), (4, 1), (3, 3), (3, 1), (2, 2), (2, 1), (1, 1)}. :
) = {(, ) | , Î, ≥}; ) = {(, ) | , Î, =2};
) = {(, ) | , Î, }; ) = {(, ) | , Î, =, ÎN}.
4. , , Î, Î. , (, )Î:
) Î; ) Î; ) (, )δ; ) (, )Ì´;
|
|
) {(, )}Ì; ) (, )Î; ) {(, )}Ì´.
4. ={1, 2, 3, 4, 5, 6} S:{(, ) | =2, , Î}. S:
) )
-2 1
-1 0 1 2 3 4 5 6
) )
0 1 2 0 1 2 3 4
-2
5. ={1, 2, 3, 4} ={2, 4, 6}. R: , Î, Î. :
) ={(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6), (4, 2), (4, 4), (4, 6)} R;
) (1, 2)δ; ) (1, 2)Î; ) (2, 2) Î;
) (4, 6)Î; ) (3, 4)Î; ) (3, 4)δ.
5. ={3, 1, 1, 2, 3, 4} R: =+2. , :
) 1R3; ) 1R3; ) 3R1; ) R(1)= 1; ) R(4)=2; ) R(2)= 1.
0. ={1, 2, 3, 4} ={5, 6, 7}, R:
R:
) R={(, )| Î, Î, £4};
) R={(1, 5), (2, 5), (2, 6), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)};
) R={(1, 5), (1, 6), (1, 7), (2, 5), (2, 6), (2, 7), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)}
: , .. , R={(1, 5), (2, 5), (2, 6), (3, 5), (3, 6), (3, 7), (4, 5), (4, 6), (4, 7)} ( ). , , £ 4, , R : R ={ (, )| Î, Î, £ 4} ( ).
0. ={, , , , } R: . :
) ={(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )} R;
) ={(, ), (, ), (, ), (, )} R;
) Ì2; ) R()=; ) R.
:
) ={(, ), (, ), (, ), (, )} R;
) R()=;
) R;
) Ì2.
( II )
1. ={4, 2, 6, 3, 5, -3, -8, -6, 0} : ) R: <; ) S: , , Î. .
1. R ={, , , } : ={(, ), (, ), (, ), (, ), (, )}. R? . R(a), R(), R(c), R(e), R-1(a), R-1(c).
2. ={2, 4, 6, 8}. . : ) : >; ) R: , , Î. .
2. Q . Q, . : Q(4), Q(-2), Q-1(1), Q-1(2).
-4 | -3 | -3 | -3 | -2 | -2 | -2 | -1 |
|
|
3. ={2, 3, 4, 6, 9, 10}. . :
) Q: , , Î.
) : , , , Î.
.
3. S ={-2, -1, 0, 1, 2, 3} ={(-2, 0), (-1, 1), (0, 2), (1, 3)}. S? . S(1), S(0), S(-1), S-1(2), S-1(3).
4. =3. , :
) ==R; ) =[-2, 2], =R; ) ={-2, -1, 0, 1, 2}, =Z.
4. , R. ? -1.
3
-2 0 1
5. >32, , : ) =R; ) =Z.
5. S, R. S? S-1(1), S-1(2).
3
-1 0 1
-2
0. ={|ÎN, ≤6} :
) R: ;
) Q: , , Î. .
:
={1, 2, 3, 4, 5, 6}
) 1 : R ={(2, 1), (4, 2), (6, 3)}; R ={(, )| , Î ; =2}.
2 . 3 :
R
4 :
2 | 4 | 6 | |
1 | 2 | 3 |
) 1 : Q={(1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (4, 2), (6, 2), (6, 3)}
Q={(, )| , Î ; = × ; ÎN}.
2 : 3 : :
0. :
-1 | -1 | -1 |
R, . R(0), R(1), R-1(1), R-1(2).
:
R R
R(0) ={1, - 1, 2},
R(1) ={-1, 1, 2},
R-1(1) ={0, 1, 2},
R-1(2) ={0, 1}
={0, 1, 2}
={-1, 1, 2}
( III )
1. ={2, 3, 4, 6, 8, 9} R: , , Î. R. , .
1. ={, , , d}
={(, ), (, ), (, ), (, ), (d, d)};
R={(, ), (, ), (, ), (, ), (, d), (d, ), (, d), (d, )};
S={(, ), (, ), (, ), (d, d), (, ), (, ), (, d), (d, )}.
:
) ; ) ; ) .
2. R {1, 2, 3, 4, 5, 6} , . . R. . , R. R-1 .
1
2 4
5 6
2. S . , ? .
|
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3. ={1, 2, 3, 4, 5, 6} R: , , Î. R. .
3. ={2, 4, 6, 8} R:≤, , Î. R. . R , , ? ?
4. , {-3, -2, -1, 1, 2}. , . . . -1, .
3
2
1
-3 -2 -1 0 1 2 3
-2
4. , , ={3, 4, 5}, . : ) {(3, 3), (3, 4), (4, 3), (4, 4), (5, 5)}; ) {(3, 3), (4, 4), (5, 5), (3, 4), (4, 5), (5, 4)}. , .
5. ={1, 2, 3, 4, 5, 6} S: , , Î. R. .
5. ={0, 1, 2, 3}
S={(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (1, 2), (0, 2), (1, 0), (2, 1), (2, 0)}.
S . ?
0. ={|ÎN, ≤4} S: , , Î. S. .
: ={1, 2, 3, 4}. S={(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)}
S-1: -
S-1={(2, 1), (3, 1), (4, 1), (3, 2), (4, 2), (4, 3)}
: - .
={(1, 1), (2, 1), (3, 1), (4, 1), (2, 2), (3, 3), (4, 4), (3, 2), (4, 2), (4, 3)}.
0. ={1, 3, 5, 7, 9} R: 2, , Î. R. . R , ? ? ?
: ={(3, 1), (5,3), (7, 5), (9, 7)} R.
R . .. , R , : , 2 . . R 1 3 .
R, , , .. , R , R (R Þ R) . .. 2, , 2. . R 1 3 .
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( IV )
1. , . :
) ( );
) ( Z);
)
( ).
1. , . , ? ?
2. , . :
) ( );
) ( N);
)
( ).
2. l. , : l, l l? ?
3. , . :
) ( Q);
)
( );
) ( N).
3. , , ? .
4. :
) ;
) , ;
) 4 , .
. .
4. . : . .
5. :
) ;
) ;
) , .
. .
5. ? .
( V )
1. Z:
) ()M3; ) 2=2; ) ||<||.
1. , . . . , ?
2. Z:
) : =4; ) ||+||= 3; ) 3=3.
2. . d?
d
3. Z:
) ||+||≠4; ) ≠; ) (22) M4.
3. , . , . , , .
4. ? : ) =Z, (-) M3; ) =N, (+) M2; ) =Z, 2=2.
4. , . , , , ? ? , , .
) )
5. ? : ) =N, (-) M3; ) = N, () M3 () M3; ) =R, =2.
5. , . , t . t, : ) tÎN, ) tÎR.
8 5
2 t
1. . . / . . , . . , . . [ .] .: , 1977. 352 .
2. - . / . . , . . - [ .]; . . . . .: , 1977. 205 .
3. . . / . . . ̳: , 1993. 350 .
4. . . : . . . / . . . ̳: , 1994. 162 .
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5. . . - / . . , . . . .: , 1985. 180 .
6. . . / . . . : , 2003. 164 .
7. . . : 2 . / . . , . . -, . . . .: , 1990. . 1. 173 .