dm1 ispit 28jan2016 reshenija
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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) )?
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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)
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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} .
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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.
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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) )?
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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))
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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} .
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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,
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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)
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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) ()() (). () ()()?
( .. ).
:
() (). : () () () () ().
, () ()().
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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.
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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 , ,
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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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