dm1 ispit 29jan2014 reshenija
TRANSCRIPT
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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
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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))
) .
. (..
.)
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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
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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
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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
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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) .
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ii) : ,
.
, .
d f, .
iii) B = {b, c, d}, infB supB.
B*={d}, supB=d
B*={a,b}, infB=b
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______________________________________________ . ________________
_____________________________________________________
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
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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
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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 .
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5. (3+10)
) :
____________
_________
__________
) A, B C :
)()( CBACBBA .
)()(
)()())(())()((
)))()((()))()(((
)))()((()))()((())((
CBACBA
CBACBACBACBABBBA
CBABBACBABBA
CBABBACBABBACBBA
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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,yM) (f(x)=f(y)x=y).
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
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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
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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) .
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ii) : , , .
, .
d f, .
iii) B = {c, d, e}, infB supB.
B*= , supB
B*={a,b,c}, infB=c
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______________________________________________ . ________________
_____________________________________________________
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
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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))
) .
. (.. .)
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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
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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,yM) (f(x)=f(y)x=y).
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
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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
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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. ( )
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) (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
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______________________________________________ . ________________
_____________________________________________________
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
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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 .
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5. (3+10)
a) :
____________
{ _____________
______________
) A, B C :
)()( CBABACA .
)()()()()(
)())(())()((
)))()((()))()(((
))(())(())((
CBACBABCABCABCA
BCABCABCAAACA
BCAACABCAACA
BACABACABACA
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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,yM) (f(x)=f(y)x=y).
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
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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. ( )
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) (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