DM1 Ispit 29Jan2014 Reshenija

8/12/2019 DM1 Ispit 29Jan2014 Reshenija

1/30

1, 29. 01. 2014

I

___________________________________ . ____________

________________________________________

:

150

1 2 3 4 5 6 7 8 9 10

1. (2+8) a) .

:

(1) .

(2) , (), (), (), () ()

.

(3) (1) (2).

)

((p q) (p r) (q r)) r

.

((p q) (p r) (q r)) r((p q) (p r) (q r)) r

((p q) ( r ( p q )))

r

((p q) ( r ( p q )))r

((p q) (r ( p q )))r((p q) (r ( p q )))r(((p q) r) ((p q) ( p q )))r(((p q) r) T)r((p q) r)r(p q) (rr)(p q) TT


2/30

2. (8) , . .

. ,

, .

?

p:

q:

r:

s:

p, , qr, (pr) s => s

1. pq

2. q

3. p Modus Tollens 1 2

4. qr

r, s

3. (3+6+3)

) ,

.

(x, y):xy .

(x)(y) (xy(x, y))

) ,

((x)( y) (xy(x, y))) (x)(y) (xy(x, y))

(x)(y) ((xy) (x, y))

(x)(y) (x=y(x, y))

(x)(y) (xy(x, y))

) .

. (..

.)


3/30

4.(2+5) a) n 4,

2, .

,

.

) n m e , n3

+ m

.

n m . k1 k2, n=2k1+1

m=2k2+1. n3+m=(2k1+1)

3+(2k2+1)=8k1

3+12k1

2+6k1+1+2k2+1=2(4k1

3+6k1

2+3k1+k2+1).

4k13

+6k12

+3k1+k2+1 , p. n3

+m=2p, n3

+m .

5. (3+10)

a) :________________

____________

{ __________

) , N P :

)())(( PNMPNNM .

)()(

)()())(())()((

)))()((()))()(((

)))()((()))()((())((

PNMPNM

PNMPNMPNMPNMNNNM

PNMNNMPNMNNM

PNMNNMPNMNNMPNNM


4/30

6. (2+3+3+2)

) .

f:ST, S

, : (xS)(yS)(xyf(x)f(y))

) f: RR f(x)=(x2+1)/(x

2+2). f ?

.

f f f .

f (x,yM) (f(x)=f(y)x=y).

f(x)=f(y) (x2+1)/(x

2+2)=(y

2+1)/(y

2+2) (x

2+1)(y

2+2)=(x

2+2)(y

2+1) x

2y

2+2x

2+y

2+2= x

2y

2+x

2+2y

2+2

x2= y

2 x=y xy

f , f .

) f: RR+ f(x)=x

2. f

-1({x|0


5/30

7. (8) )31(mod10010 41572

8 .

100102=1810, 1578=11110, 314=1310

18111

(mod 13)

185(mod 13)

5225(mod 13)

52-1(mod 13)

1825

2(mod 13)

182-1(mod 13)

(182)

55(-1)

55(mod 13)

18110

-1(mod 13)

18*18110

5*(-1)(mod 13)

18111-5(mod 13)18

1118(mod 13)

8. (8)

22

1

4

0

3*21

i

i

i

i

.

253

243

23*11

243

23*11

2

242*21

23*11

2

)1243(*21

2

23*22

)13

13(*213*21

5

22

1

4

0

i

i

i

i


6/30

9. (9) 2 - 27 + 272

- . + 2(-7)n = o

n.

.

: n=0

4

8

4

71

4

)7(1

4

)7(12

10

P(0) .

: a n=k

2 - 27 + 272

- . + 2(-7)k =

4

)7(1 1 k

n=k+1

2 - 27 + 272

- . + 2(-7)k+2(-7)

k+1=

4

)7(1 1

k

+2(-7)k+1

=

4

)7(1

4

)7)(7(1

4

)7(8)7(1 2111

kkkk

P(k)P(k+1),

.

10.a) (2+2+2) "".

:

i) .

, ,

(xA) x a.

ii) .

,

,

(xA) a xa = x.

iii) sup B, B .

v B,

B. ( )

) (4+3+2) A = {a,b,c,d,e,f} o ""

:

= A{(a, b), (a,c), (a,d), (a, f), (b, c), (b,d), (b, f), (c, d), (c, f), (e, f)}.

i) .


7/30

ii) : ,

.

, .

d f, .

iii) B = {b, c, d}, infB supB.

B*={d}, supB=d

B*={a,b}, infB=b


8/30

______________________________________________ . ________________

_____________________________________________________

1, 29.01.2014

II

:

150

1 2 3 4 5 6 7 8 9 10

1. (2+8) a) .

, .

)

((p q) (p r)) (q r) .

((p q) (p r)) (q r)((p q) (p r)) (q r)

((p q) (p r)) (q r) ((p q) (p r)) (q r)

((p p) (p r) (q p) (q r)) (q r)

(T (p r) (q p) (q r)) (q r)

((p r) (q p) (q r)) (q r)(p r q r) (q p q r) (q r q r)

(p q r r) (q q r p) (q q r r)

(p q r r) (q q r p) (q q r r)(p q T) (T r p) (T T)T T TT


9/30

2. (8) .

. .

. ?

p:

q:

r:

s:

p

q, q

r, (sp)

, p => s

1. p

2. pq

3. q Modus Ponens 1 2

4. q r

5. r Modus Ponens 3 4

6. (sp) r

7. (sp) Modus Tollens 5 6

8. s p 7

9. s 8


10/30

3. (3+6+3) Facebook.

) ,

Facebook.

(x, y):xy .

(x)(y) (xy(x, y))

) ,

((x)( y) (xy(x, y))) (x)(y) (xy(x, y))

(x)(y) ((xy) (x, y))

(x)(y) (x=y(x, y))

(x)(y) (xy(x, y))

) .

Facebook. (.. e

Facebook.)

4.(2+5) a) n 6,

2., .

,

.

) n e m e , n+m3

.

ne me . k1 k2,

n=2k1 m=2k2+1. n+m3=2k1+(2k2+1)

3=2k1+8k2

3+12k2

2+6k2+1=2(k1+4k2

3+6k2

2+3k2)+1.

k1+4k23+6k2

2+3k2 , p. n+m

3=2p+1, n+m

3 .


11/30

5. (3+10)

) :

____________

_________

__________

) A, B C :

)()( CBACBBA .

)()(

)()())(())()((

)))()((()))()(((

)))()((()))()((())((

CBACBA

CBACBACBACBABBBA

CBABBACBABBA

CBABBACBABBACBBA


12/30

6.(2+3+3+2)

) .

f:ST yT

xS f(x) = y, (y)(xS)(f(x) = y)

) f: RR f(x)=(x2+2)/(x

2+3). f ?

.

f f f .


f(x)=f(y) (x2+2)/(x

2+3)=(y

2+2)/(y

2+3) (x

2+2)(y

2+3)=(x

2+3)(y

2+2) x

2y

2+3x

2+2y

2+6= x

2y

2+2x

2+3y

2+6

x2= y

2 x=y xy

f , f .

) f: RR+ f(x)=x

2. f

-1({x|x>4}).

f-1

({x|x>4}) = {x|x2}

) f ) ? .

f (x,yR) (x y f(x) f(y)), a f e (x,yR) (x y

f(x) f(y)) . , x


13/30

7. (8)

101112=2310, 5916=8910, 1014=1710

2389

(mod 17)

236(mod 17)

6236(mod 17)

622(mod 17)

2416(mod 17)

24-1(mod 17)

(62)

4-1(mod 17)

68-1(mod 17)

238-1(mod 17)

(238)11(-1)11(mod 17)

2388

-1(mod 17)

23*2388

6*(-1)(mod 17)

2389

-6(mod 17)

2389

11(mod 17)

8. (8)

9

26

36*6

4*13*12

4

36*256

25*13*12

4

6*5

6

25*13*12

225

1

3

12

1

2

i

i

i

i


14/30

9.(9) 123 + 234 + ... + n(n+1)(n+2) =

o n.

.

: n=1

123 = 6 =4

4321 P(0) .

: a n=k

123 + 234 + ... + k(k+1)(k+2) =4

)3)(2)(1( kkkk

n=k+1123 + 234 + ... + k(k+1)(k+2) + (k+1)(k+2)(k+3) =

4

)3)(2)(1( kkkk+(k+1)(k+2)(k+3) =

4

)4)(3)(2)(1(

4

)3)(2)(1(4)3)(2)(1(

kkkkkkkkkkk

P(k)P(k+1),

.

10. a) (2+2+2) "".

i)

,

,

(xA) x.

ii)

,

, ,

(xA) x x = .

iii) B , inf B.

u B,

B. ( )

) (4+3+2) A={a,b,c,d,e,f}, , o ""

:

= A{(a, b), (a,c), (a,d), (a, ), (a, f), (b,c), (b, ), (b, f), (c, d), (c, e), (c, f), (e, f)}.

i) .


15/30

ii) : , , .

, .

d f, .

iii) B = {c, d, e}, infB supB.

B*= , supB

B*={a,b,c}, infB=c


16/30

______________________________________________ . ________________

_____________________________________________________

1, 29.01.2014

III

:

150

1 2 3 4 5 6 7 8 9 10

1. (2+8) a) .

, .

)

((r p) (r q) (p q)) q

.

((r p) (r q) (p q)) q((r p) (r q) (p q)) q((r p) ( q ( r p ))) q((r p) ( q ( r p )))q((r p) (q ( r p )))q((r p) (q ( r p )))q(((r p) q) ((r p) ( r p )))q(((r p) q) T)q((r p) q)q(r p) (qq)(r p) TT


17/30

2. (8) .

. .

.

?

p:

q:

r:

s:

p, , qr, (pr) s => s

p, , qr, (pr) s => s

1. pq

2. q

3. p Modus Tollens 1 2

4. qr

r

3. (3+6+3) .

) ,

.

(x, y):xy .

(x)(y) (xy(x, y))

) ,

((x)( y) (xy(x, y))) (x)(y) (xy(x, y))

(x)(y) ((xy) (x, y))

(x)(y) (x=y(x, y))

(x)(y) (xy(x, y))

) .

. (.. .)


18/30

4.(2+5) a) n 8,

4., .

,

.

) n e m ,

(n+m)2 .

ne me . k1 k2,

n=2k1+1 m=2k2.

(n+m)2

=(2k1+1+2k2)2

=4k12

+4k22

+4k1k2+4k1+4k2+1=2(2k12

+2k22

+2k1k2+2k1+2k2)+1.

2k12+2k2

2+2k1k2+2k1+2k2 , p. (n+m)

2=2p+1, (n+m)

2 .

5. (3+10)

a) :

___________

____________

{ _________

) , N P :

)())(( PNMNMPM

)()()()()(

)())(())()((

)))()((()))()(((

))(())(())((

PNMPNMNPMNPMNPM

NPMNPMNPMMMPM

NPMMPMNPMMPM

NMPMNMPMNMPM


19/30

6.(2+3+3+2)

) .

f:ST, S

, : (xS)(yS)(xyf(x)f(y))

) f: RR f(x)=(x2+3)/(x

2+4). f ?

.

f f f .


f(x)=f(y) (x2+3)/(x

2+4)=(y

2+3)/(y

2+4) (x

2+3)(y

2+4)=(x

2+4)(y

2+3) x

2y

2+4x

2+3y

2+12= x

2y

2+3x

2+4y

2+12

x2= y

2 x=y xy

f , f .

) f: RR+ f(x)=x

2. f

-1({x|x>2}).

f-1

({x|x>2}) = {x| 2x } U {x| x2 }

) f ) ? .

f (x,yR) (x y f(x) f(y)), a f e (x,yR) (x y

f(x) f(y)) . , x


20/30

7. (8)

1024=1810, 6F16=11110, 11012=1310

18111

(mod 13)

185(mod 13)

5225(mod 13)

52-1(mod 13)

1825

2(mod 13)

182-1(mod 13)

(182)

55(-1)

55(mod 13)

18110

-1(mod 13)

18*181105*(-1)(mod 13)

18111

-5(mod 13)

18111

8(mod 13)

8. (8)

5

66

5

6*11

30

36*11

11*30

36*121

2

11*10*6

4

144*121

*610

1

11

1

3

i

i

i

i


21/30

9. (9) 14+ 2

4+ ... + n

4= o

n.

.

: n=1

14= 1 =

30

5321 P(0) .

: a n=k

14+ 2

4+ ... +k

4=

30

)133)(12)(1( 2

kkkkk

n=k+1

14+ 2

4+ ... +k

4+(k+1)

4=

30

)133)(12)(1( 2 kkkkk+(k+1)

4=

30

)1(30)133)(12)(1( 42 kkkkkk

30

))1(30)133)(12()(1( 32 kkkkkk

30

)308991396)(1( 234

kkkkk

30

)1)1(3)1(3)(1)1(2)(2)(1( 2

kkkkk

P(k)P(k+1), .

10.a) (2+2+2) "".

i)

,

, ,

(xA) x x = .

ii)

,,

(xA) x.

iii) B , inf B.

u B,

B. ( )


22/30

) (4+3+2) A={a,b,c,d,e,f}, , o ""

:= A{(a, b), (a,c), (a,e), (a, f), (b,c), (b, ), (b, f), (c, e), (c, f), (e, f)}.

i) .

ii) : , , .

d, .

d f, .

iii) B = {a, d, e}, infB supB.

B*= , supB

B*= , infB


23/30

______________________________________________ . ________________

_____________________________________________________

1, 29.01.2014

IV

:

150

1 2 3 4 5 6 7 8 9 10

1. (2+8) a) () .

()

.

)

((r p) (r q)) (p q)

.

((r p) (r q)) (p q)((r p) (r q)) (p q)

((r p) (r q)) (p q) ((r p) (r q)) (p q)

((r r) (r q) (p r) (p q)) (p q)

(T (r q) (p r) (p q)) (p q)

((r q) (p r) (p q)) (p q)(r q p q) (p r p q) (p q p q)

(r p q q) (p p q r) (p p q q)

(r p q q) (p p q r) (p p q q)(r p T) (T q r) (T T)T T TT


24/30


25/30

3. (3+6+3)

.) ,

.

(x, y):xy .

(x)(y) (xy(x, y))

) ,

((x)( y) (xy(x, y))) (x)(y) (xy(x, y))

(x)(y) ((xy) (x, y))(x)(y) (x=y(x, y))

(x)(y) (xy(x, y))

) .

. (..

.)

4.(2+5) a) n 6,

3., .

,

.

) n e m ,

(n+m)2 .

ne me . k1 k2,

n=2k1 m=2k2+1.

(n+m)2=(2k1+2k2+1)

2=4k1

2+4k2

2+4k1k2+4k1+4k2+1=2(2k1

2+2k2

2+2k1k2+2k1+2k2)+1.

2k12+2k22+2k1k2+2k1+2k2 , p. (n+m)2=2p+1, (n+m)2 .


26/30

5. (3+10)

a) :

____________

{ _____________

______________

) A, B C :

)()( CBABACA .

)()()()()(

)())(())()((

)))()((()))()(((

))(())(())((

CBACBABCABCABCA

BCABCABCAAACA

BCAACABCAACA

BACABACABACA


27/30

6.(2+3+3+2)

) .

f:ST yT

xS f(x) = y, (y)(xS)(f(x) = y)

) f: RR f(x)=(x2+4)/(x

2+5). f ?

.

f f f .


f(x)=f(y) (x2+4)/(x

2+5)=(y

2+4)/(y

2+5) (x

2+4)(y

2+5)=(x

2+4)(y

2+5) x

2y

2+5x

2+4y

2+20= x

2y

2+4x

2+5y

2+20

x2= y

2 x=y xy

f , f .

) f: RR+ f(x)=x

2. f

-1({x|2


28/30


29/30

9. (9) 12- 2

2+ 3

2- ... + (-1)

n-1n

2= o

n.

.

: n=1

2

2

2

21)1(11

112

P(0) .

: a n=k

12- 2

2+ 3

2- ... + (-1)

k-1k

2=

2

)1()1( 1

kkk

n=k+1

12- 2

2+ 3

2- ... + (-1)

k-1k

2+(-1)

k(k+1)

2=

2

)1()1( 1

kkk

+(-1)k(k+1)

2=

2

)1()1(2)1()1( 21

kkk kk

=

2

))1)(1(2)(1()1( 1

kkkk

=

2

)22)(1()1( 1

kkkk

=

2

)2)(1()1( 1

kkk

=

2

)2)(1()1( kkk

P(k)P(k+1),

.

10.a) (2+2+2) "".

i) ,

,

(xA), a xa = x.

ii)

, ,

(xA), x a.

iii) B , sup B.

v B, B. ( )


30/30

) (4+3+2) A={a,b,c,d,e,f}, , o ""

:

= A{(a, b), (a,c), (a,d), (a, e), (a, f), (b,d), (b, ), (b, f), (c, d), (c, e), (c, f), (d, f)}.

i) .

ii) : , , .

, .

f, .

iii) B = {a, c, d}, infB supB.

B*={d,f}, supB =d

B*={a}, infB=a

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DM1 Ispit 29Jan2014 Reshenija