Спец_главы_математики

Вариант 4

1.а) y; :y; x ` (x _ y) ^ z; б) x ! y; y ! z ` x ! z;

в) x ^ :y; x ^ y ` x ! (y ! x); г) (x ^ y) _ z ` (x _ z) ^ x.

2.A = hZ; ·i; B = hN; ·i:

3.9x9y P (x; y) ! 9y9x P (x; y):

4.9x8y P (x; y) ^ :8x9x P (x; y):

5.8x(9yP (x; y) ! Q(x; y)) ! (:8x9yQ(x; y) ^ 9xR(x; y)):

6.Á1 = 9x8y9z8u(P3(y; x) ! (P1(y; u) _ P2(x; z))),

Á2 = 9x8y9z8u(P4(y) _ (P3(y; u) ^ (P1(x; z) ! P2(y; u)))).

7.f(x; y) = x ¢ y + 1:

141

Вариант 5

1.а) ` x ! ::x;

б) ` ((x ! y) ! y) ! x;

в) x; y _ z; u ` (x ^ y) ! ((z _ x) ^ u); г) x; u _ y ` (z ! x) _ (:z ! u).

2.A = hR; ¢i; B = hN; ¢i:

3.8x8y P (x; y) ! 8x P (x; x):

4.8x8y9z(P (x; z) ^ P (z; y) & :(x = z) ^ :(z = y)):

5.8x9y:(P (x; y) ! :Q(x; y)) _ :9x8y R(x; y):

6.Á1 = 9x8y8z9u8v(:P2(y; x) ^ :(f2(y; z) = f3(u; v))), Á2 = 8x8y8z(P1(f1(x); x; y) ! (f2(x; y) = f3(z; x))),

Á3 = 8x8y8z8u(P1(x; y; z)_(:P2(f1(x); u) ! (f2(y; z) = f3(u; v)))).

7. f(x) = ½	x	1; при x	= 0
	0;¡	при x	6= 0:

142

Вариант 6

1.а) x _ y ` (x ^ y) ! (x _ z);

б) x; y _ u; z _ y ` (y ! x) _ (u ! y); в) ` (x ! y) _ (y ! x);

г) x ! y; y ! z ` (x ^ z) ! y.

2.A = hN; +i; B = hZ; +i:

3.9x P (x; x) ! 9x9y P (x; y):

4.:8x8y(x = y) ^ 8x8y(P (x; y) _ P (y; x)):

5.:8x9y (P (x; y) ^ :9x8yQ(x; y):

6.Á1 = 9x8y9z8u:(P3(y; u) ! P1(x; x; z)),

Á2 = 9x8y8z8u8v((:P1(y; z; u) ^ P3(y; z)) ! P2(y; x; u; v)).

7.f(x) = 2x + 1:

143

Вариант 7

1.а) x ! y ` :y ! :x;

б) x ! y ` (x _ z) ! (y _ z); в) x ! (y ! z) ` (x ^ y) ! z;

г) x ^ y; y _ z; z _ :u ` (x ! u) _ (z ! y).

2.A = hN; ¢i; B = hZ; ¢i:

3.9x P (x) ! Q(x)) $ (8x P (x) ! 9xQ(x)):

4.9x9y(:(x = y) ^ 8x8y8z(P (x; y) ^ P (y; z) ! P (x; z)):

5.8x P (x; y) ! :9y9x Q(x; y):

6.Á1 = 8x8y9z:P (z; x; y),

Á2 = 8x8y8z8u8v8w((P (x; y; u)^P (x; v; w)) ! (P (y; z; v)_P (u; z; w))), Á3 = 8x9z8y:(P (x; f(x; y); y) _ P (z; x; z)).

7.f(x) = x + 6:

144

Вариант 8

1.а) x ! y ` (z ! x) ! (z ! y); б) x ! y ` (x ^ z) ! (y ^ z);

в) x _ y; (y _ :z) _ u ` (z ! x) ^ (x _ u); г) x ` y ! x.

2.A = hR; +i; B = hN; ¢i:

3.(P (x) ! 9yQ(y)) $ 9y(P (x) ! Q(y)):

4.9x9y P (x; y) ^ 8x8y(P (x; y) ^ P (y; x) ! :(x = y)):

5.8x (9y P (x; y) _ 8yQ(x; y)) ^ 9x:(9yP (x; y) _ 8yQ(x; y)):

6.Á1 = 8y(P1(y) ! (P2(y) _ 9xP3(x; y))),

Á2 = :(8x((9y(P1(y) ^ P3(x; y)) ! 9z(P2(z) ^ P3(x; z)))).

7.f(x) = 2x:

145

Вариант 9

1.а) :x ! :y ` y ! x; б) :(x ^ :y) ` :x _ y;

в) x _ y; x _ z; x _ :z _ u ` u _ y; г) x ! (y _ z) ` (x ^ :y) ! z.

2.A = hN; +i; B = hZ; +i:

3.(8x P (x) ! Q(y)) $ 9x(P (x) ! Q(y)):

4.9x8y(P (x; y) ! 8z:R(x; y; z)) ^ 9x9y P (x; y):

5.8x(:(9yP (x; y) ! 8yQ(x; y)) _ (9yP (x; y) ! 8yQ(x; y))):

6.Á1 = 8x(9y(P1(x; y) ^ P2(y)) ! 9y(P3(y) ^ P4(x; y))), Á2 = :(:9xP3(x) ! 8x8yP1(x; y) ! :P2(y))).

7.f(x) = 2x + 2:

146

Вариант 10

1.а) x ! (:z _ y) ` (x ^ z) ! y;

б) x _ y; z _ :y _ u ` (x ^ y) _ (u ! z); в) x ! y; y ! :x `;

г) (x ^ y) ! (x _ u) ` u ! (x _ y).

2.A = hR; ¢i; B = hZ; ¢i:

3.(9x P (x) ! Q(y)) $ 8x(P (x) ! Q(y)):

4.8x P (x; x) ^ 8x9y(P (x; y) ^ :(x = y)):

5.(:9x9yP (x; y) _ 9x8yQ(x; y)) ^ :8x9yR(x; y):

6.Á1 = 8x((P1(x) ^ :P2(x)) ! 9y(P3(x; y) ^ P4(y))),

Á2 = 9x(P5(x) ^ P4(x) ^ 8y(P3(x; y) ! P5(y))),

Á3 = 8x(P5(x) ! :P2(x)).

7.f(x) = x + 5:

147

Вариант 11

1.а) x _ y; x ! y; y ! z ` u _ z; б) ` (x ^ y) ! (x _ y);

в) x _ y; :x _ z; u ^ (x ! z) ` (u ^ y) ! z; г) y ` (y ! x) ! x.

2.A = hC; +i; B = hN; +i:

3.(P (x) ! 8yQ(y)) $ 8y(P (x) ! Q(y)):

4.9x:(f(x) = x) ^ 9x(f(f(x)) = x):

5.:((9x9yP (x; y) ^ 9x8yQ(x; y))) _ 9x9yR(x; y):

6.Á1 = 8x8y9z((P1(x; y) ! P2(x; y)) ^ P3(x; y; z)), Á2 = 8x8y(P2(x; y) ! P3(x; y)),

Á3 = 9x9y9z(P1(x; y) ^ :P3(x; y; z)).

7.f(x; y) = x + y + 1:

148

Вариант 12

1.а) x ! (y ! z) ` (x ! y) ! z; б) x ! y ` (x _ z) ! (y _ z);

в) x ^ (y _ z); :y _ z; x _ :z _ y ` y; г) x ! y ` :(x ^ :y).

2.A = hC; +i; B = hZ; +i:

3.(8xP (x) _ 8xQ(x)) ! 8x(P (x) _ Q(x)):

4.8x(P (x) ! Q(x)) ^ 9x9y(:(x = y) ^ :P (x) ^ :Q(x) ^ :P (y) ^ :Q(y)):

5.9x(9yP (x; y) _ 8yQ(x; y)) ! 9y8y(P (x; y) ^ Q(x; y)):

6.Á1 = 9x8y9z(P1(x; y) ^ :P2(x; z) ^ P3(x; y; z)), Á2 = 8x8y(P1(x; y) ! P2(x; y)),

Á3 = 8x9y8z(:P3(x; y; z) ^ :P1(x; y)).

7. f(x; y) = ½	x	y; при x	y
	0;¡	при x	¸< y:

149

Вариант 13

1.а) ` (x ! z) _ (z ! x);

б) y; x _ u ` (z ! y) _ (:z ! y); в) ` ::x ! x;

г) x _ y; x _ :y _ u; y _ u ` :y _ u.

2.A = hC; +i; B = hR; +i:

3.9x(P (x) ^ Q(x)) ! (9xP (x) ^ 9xQ(x)):

4.8x(:P (x; x) ^ :P (x; s(x)) ^ P (x; s(s(x)))):

5.9x8yP (x; y) ! :(8x:(8yP (x; y) ^ 9yQ(x; y))):

6.Á1 = 8x9y(f(x) = f(f(y)) ^ P1(x; y)), Á2 = 9x8y8z(P2(x; y) ^ P3(x; y; z)), Á3 = 9x8y8z(P1(x; y) ! P3(x; y; z)).

7.f(x; y) = x ¢ y + 2:

150