1, 28. 01. 2016

I

___________________________________ . ____________

________________________________________

: 150

1 2 3 4 5 6 7 8 9 10

1. (18)

) (2)

:

, .

) (6) :

i) p___________ ii) p(pr) _____p_______ ii) (pq) __ p q______

) (8) (

4 )

(pq)r(pr)q

:

(pq)r p q r ( p r) q (pr)q

) (2) )?

2. (10) q, qr p(qr) p?

:

1. q ()

2. qr ()

3. p(qr) ()

4. r (.. 1 2)

5. qr ( 1 4)

6. (qr) ( 5)

7. p (.. 3 6)

3. (10) f: AB p:

,

q: , .

pq.

:

p: (xA) (yA)(x y f(x)=f(y))

q: (xA) (yA)(f(x)=f(y) x = y)

((xA) (yA)(x y f(x)=f(y))) (xA) (yA) (x y f(x)=f(y))

(xA) (yA) ((f(x) = f(y)) x = y) (xA) (yA)(f(x)=f(y) x = y)

4. (16) ) (3) .

: ={(x, y)| xA yB}

) (2) ={1, 3} B={2, 4}.

: ={(1, 2), (1, 4), (3, 2), (3, 4)}

) (3) :

i) A () ________ ii) A () ___ T______ iii) {A} () ___ T______

( 3 0 )

) (8) () ()(), ()() ()

( .. ).

:

() (). :

()() () () ( ) ().

={1} ={2}. () = ({1, 2}) ={, {1}, {2}, {1, 2}}, a () ()= ({1}) ({2}) = {, {1}}{, {2}}={, {1}, {2}}. () () (), {1, 2} .

5. (5) a, b, c . (a | b b | c ) (a | c).

: b = k1a c=k2b, 1 2 Z c=k2k1a a|c 6. (8) S:

If (x0. y:= 2|x|/x |x|=x y=2x/x=2.

7. (10) 11!+22!+33!+...+nn!=(n+1)! 1, n>0.

:

n=1

11! = (1+1)! 1 : n=k , ..

11!+22!+33!+...+kk!=(k+1)! 1 n=k+1, :

11!+22!+33!+...+kk!+(k+1)(k+1)!= (k+1)! 1+(k+1)(k+1)! = (k+1)! +(k+1)(k+1)! 1 = (k+1)! (1+(k+1))-1= (k+1)! (k+2))-1=(k+2)! 1

8. (4) f: AB, A={1, 2, 3} B={1, 2, 3, 4}, : (1 2 34 3 1

).

g: BA, g f = 1A. :

: (1 2 3 43 3 2 1

)

9. (13)

a) 1. , ,

. ( ,

, )

i) (2) : R1

ii) (2) : R2, R3

iii) (2) : R1, R2

iv) (2) : R1, R2, R3

1

) (5) R = {1, 2, 3, 4, 5, 6}

A/R={{1, 2, 3}, {4, 6}, {5}}. !

:

=

[ 1 1 1 0 0 01 1 1 0 0 01 1 1 0 0 00 0 0 1 0 10 0 0 0 1 00 0 0 1 0 1]

10. (6)

R .

) (3) ? ( )

i) .____ _______

ii) .____________

iii) Sup(A), , . ___________

) (3) iii

: , ={1,2,3}. R aRb a|b.

Sup(A)=6, .

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1, 28. 01. 2016

II

___________________________________ . ____________

________________________________________

: 150

1 2 3 4 5 6 7 8 9 10

1. (18)

) (2)

:

, .

) (6) :

i) p______p_____ ii) (qr)p __(qp)(rp)___ ii) (pq) ___pq____

) (8) (

4 )

q(pr) r (pq) :

1. (q(pr)) = , (r (pq)) =

(q(pr)) = (q) = (pr) = .. (p) = (r) = .

r(pq) () , 2. (q(pr)) = , (r (pq)) =

(r (pq)) = (r) = (pq) = .. (p) = (q) = .

q(pr) () ,

) (2) )?

2. (10) q, qr (qr)p p?

:

1. q ()

2. qr ()

3. (qr)p ()

4. r (.. 1 2)

5. qr ( 1 4)

6. ( qr) ( 5)

7. p (.. 3 6)

3. (10) f: AB p:

,

q: , .

pq.

:

p: (xA) (yA)(x y f(x)=f(y))

q: (xA) (yA)(f(x)=f(y) x = y)

((xA) (yA)(f(x)=f(y) x = y)) (xA)(yA) (f(x)=f(y) x = y)

(xA)(yA) ((f(x)=f(y)) x = y) (xA)(yA)(f(x)=f(y) (x = y))

(xA) (yA)(x y f(x)=f(y))

4. (16) ) (3) .

: () ={X | XA }

) (2) ={1, 2, 3}.

:() ={, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

) (3) :

i) () ____T____ ii) () ____T____ iii) {} () __T______

( 3 0 )

) (8) ()() (), () ()()

( .. ).

:

() (). :

()() () () ( ) ().

={1} ={2}. () = ({1, 2}) ={, {1}, {2}, {1, 2}}, a () ()= ({1}) ({2}) = {, {1}}{, {2}}={, {1}, {2}}. () () (), {1, 2} .

5. (5) a, b, c . (a | c b | d ) (ab | cd).

: c = k1a d=k2b, 1 2 Z cd=k2k1ab ab|cd 6. (8) S:

If (x>0) then {y:= 2|x|/x} else { y:= -2|x|/x }

p{S}q, p: x0, q:y=2.

:

x > 0 y:= 2|x|/x |x| e x y=2x/x=2. ( > 0)

0, p: x0,

7. (10) 3 + 35 + 352 +. . . + 35n =3(5n+ l - 1)/4, n0.

:

n=0 3 = 3(5l - 1)/4

: n=k , .. 3 + 35 + 352 +. . . + 35k =3(5k+ l - 1)/4

n=k+1, : 3 + 35 + 352 +. . . + 35k + 35k+1 = 3(5k+ l - 1)/4 + 35k+1 = (35k+1 3 + 435k+1)/4 = (155k+1-3)/4 = 3 (55k+1-1)/4 = 3 (5k+2-1)/4

8. (4) g: BA, A={1, 2, 3} B={1, 2, 3, 4}, : (1 2 3 43 1 2 1

).

f: AB, g f = 1A. :

: (1 2 32 3 1

)

9. (13)

a) 1. , ,

. ( ,

, )

i) (2) : R2

ii) (2) : R1, R2

iii) (2) : R3

iv) (2) : R1, R2

1

) (5) R = {1, 2, 3, 4, 5, 6}

A/R={{1, 2}, {3, 4, 6}, {5}}. !

:

=

[ 1 1 0 0 0 01 1 1 0 0 00 0 1 1 0 10 0 1 1 0 10 0 0 0 1 00 0 1 1 0 1]

10. (6) R .

) (3) ? ( )

i) ._____ _______

ii) .______T______

iii) inf(A), , . __________

) (3) iii

: , ={2,3,4}. R aRb a|b.

inf(A)=1, .

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1, 28. 01. 2016

III

___________________________________ . ____________

________________________________________

: 150

1 2 3 4 5 6 7 8 9 10

1. (18)

) (2)

:

()

.

) (6) :

i) p___________ ii) p(qr) ____(pq)( pr)________ ii) (pq) ____pq___

) (8) (

4 )

(pq)r(pr)q

:

(p) = , (q)r(r)q qrrq

(p) = , (q)r(r)q rq

) (2) )?

2. (10) q, rq (pq)r p?

:

1. q ()

2. rq ()

3. (pq)r ()

4. r (.. 1 2)

5. (pq) (.. 3 4)

6. pq ( 5)

7. p ( 6)

3. (10) f: AB p:

B ,

q: .

pq.

:

p: (yB)(xA)(f(x) y)

q: (yB) (xA)(f(x) = y)

((yB)(xA)(f(x) y)) (yB) (xA)(f(x) y) (yB) (xA)(f(x) = y)

4. (16) ) (3) .

: () ={X | XA }

) (2) B={a, b, c}.

:() ={, {a}, { b }, { c }, { a, b }, { a, c }, { b, c }, { a, b, c }}

) (3) B :

i) B (B) ________ ii) B (B) ___T_____ iii) {B} (B) ____T____

( 3 0 )

) (8) ()() (). () ()()?

( .. ).

:

() (). : () () () () ().

, () ()().

5. (5) a, b, c . (a | b a | c ) (a | bc).

: b = k1a c=k2a, 1 2 Z bc=k2k1aa a|cd 6. (8) S:

If (x0. y:= 3|x|/x |x|=x y=3x/x=3.

7. (10) 2 - 27 + 272 - . . . + 2 (-7)n = (1 - (-7)n+1)/4, n0.

:

n=0 2 = (1 - (-7)1)/4

: n=k , ..

2 - 27 + 272 - . . . + 2 (-7)k = (1 - (-7)k+1)/4 n=k+1, :

2 - 27 + 272 - . . . + 2 (-7)k + 2 (-7)k+1 = (1 - (-7)k+1)/4+ 2 (-7)k+1= (1 - (-7)k+1 +8 (-7)k+1)/4 = = (1 +7(-7)k+1)/4 = (1 - (-7)k+2)/4

8. (4) f: AB, A={1, 2, 3, 4} B={1, 2, 3}, : (1 2 3 42 3 1 2

).

g: BA, f g= 1B. :

: (1 2 33 1 2

)

9. (13)

a) 1. , ,

. ( ,

, )

i) (2) : R2

ii) (2) : R1, R2

iii) (2) : R2, R3

iv) (2) : R1, R2

1

) (5) R = {1, 2, 3, 4, 5, 6}

A/R={{1}, {2, 3, 4}, {5, 6}}. !

:

=

[ 1 0 0 0 0 00 1 1 1 0 00 1 1 1 0 00 1 1 1 0 00 0 0 0 1 10 0 0 0 1 1]

10. (6) R .

) (3) ? ( )

i) A , .____T_______

ii) A , ._____ _______

iii) A , ._____T______

) (3) ii

: ={1,2,3}, A={1, 2} R={(1, 3), (2,3)}. 3 , ,

--- ---

1, 28. 01. 2016

IV

___________________________________ . ____________

________________________________________

: 150

1 2 3 4 5 6 7 8 9 10

1. (18)

) (2)

:

, .

) (6) :

i) p ___ p ______ ii) (pr)p _____p______ ii) (pq) ____pq ____

) (8) (

4 )

r(pq)q(pr)

:

r(pq) r(pq) rpq (rp)q (p r)q q(p r)

) (2) )?

2. (10) q, qr q(pr) p?

:

1. q ()

2. qr ()

3. q(pr) ()

4. r (.. 1 2)

5. (pr) (.. 1 3)

6. pr ( 5)

7. p ( 4 6)

3. (10) f: AB p:

B ,

q: .

pq.

:

p: (yB) (xA) (f(x) y)

q: (yB) (xA)(f(x) = y)

((yB) (xA)(f(x) = y)) (yB)(xA)(f(x) = y) (yB)(xA)(f(x) y)

4. (16) ) (3) .

: ={(x, y)| xA yB}

) (2) ={1, 2} B={3, 4}.

: ={(1, 3), (1, 4), (2, 3), (2, 4)}

) (3) B :

i) (B) ____T____ ii) (B) ____T____ iii) {} (B) ___T_____

( 3 0 )

) (8) () ()(). ()() ()?

( .. ).

( ). : () () () () ()

, ()() ().

5. (5) a, b, c . (a | b ab | c ) (a | c).

: b = k1a c=k2ab, 1 2 Z c=k2k1aa a|c 6. (8) S:

If (x>0) then {y:= 3x/|x|} else { y:= -3x/|x| }

p{S}q, p: x0, q:y=3.

:

x > 0 y:= 3|x|/x |x| e x y=3x/x=3. ( > 0)

0, p: x0,

7. (10) 2 + 222 + 323 + ... + n2n = (n-1) 2n+1 + 2, n>0.

:

n=1

2 = (1-1) 21+1 + 2 : n=k , ..

2 + 222 + 323 + ... + k2k = (k-1) 2k+1 + 2 n=k+1, :

2 + 222 + 323 + ... + k2k + (k+1)2k+1 = (k-1) 2k+1 + 2 + (k+1)2k+1 = (k-1+k+1) 2k+1 + 2 = 2k2k+1 + 2 =

k2k+2 + 2

8. (4) g: BA, A={1, 2, 3, 4} B={1, 2, 3}, : (1 2 33 2 1

).

f: AB, fg = 1B. :

: (1 2 3 43 2 1 1

)

9. (13)

a) 1. , ,

. ( ,

, )

i) (2) : R2

ii) (2) : R1, R2, R3

iii) (2) : R2

iv) (2) : R1, R2, R3

1

) (5) R = {1, 2, 3, 4, 5, 6}

A/R={{1, 2},{3, 4, 5},{6}}. !

:

=

[ 1 1 0 0 0 01 1 0 0 0 00 0 1 1 1 00 0 1 1 1 00 0 1 1 1 00 0 0 0 0 1]

10. (6) R .

) (3) ? ( )

i) A , .____T________

ii) A , ._____ _______

iii) A , .______T____

) (3) ii

: ={1,2,3}, A={2, 3} R={(1, 2), (1, 3)}. 1 ,

, .

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