Дискретная математика Насоров А.З., Насыров З.Х. 2009
.pdf
kontrolxnye zadaniq
nIVE PRIWODQTSQ 8 KONTROLXNYH ZADA^ PO KURSU DISKRETNOJ MATEMATIKI. kAVDAQ ZADA^A PREDLAGAETSQ W 20 WARIANTAH. nOMER SWOEGO WARIANTA STUDENTY ZAO^NOJ FORMY OBU^ENIQ MOGUT OPRE- DELITX SAMOSTOQTELXNO. dLQ \TOGO NADA NAJTI OSTATOK OT DELENIQ NOMERA SWOEJ ZA^ETNOJ KNIVKI NA ^ISLO 20. nAPRIMER, ESLI NOMER ZA^ETNOJ KNIVKI OKAN^IWAETSQ NA 37, TO NADO WYPOLNQTX WARIANT 17, A, ESLI \TOT NOMER OKAN^IWAETSQ NA 40, TO WYPOLNQETSQ WARI- ANT 0.
1. nAJDITE MNOGO^LEN vEGALKINA, MINIMALXNU@ dnf I POSTROJ- TE KONTAKTNU@ SHEMU DLQ FUNKCII f(a; b; c; d), ZADANNOJ STOLBCOM ZNA^ENIJ W STANDARTNOJ TABLICE ISTINNOSTI
0) f = (0; 1; 4; 5; 8 ; 11; 13), |
1) f = (0; 2; 4 |
; 8; 10; 14), |
|||
2) f = (1; 3 ; 9; 11;12), |
3) f = (2 ; 4;6; 8 ; 13), |
||||
4) f = (1 |
; 6; 9; 12 ; 14), |
5) f = (0; 2; 4; 5; 8;10; 12 ; 14), |
|||
6) f = (0 |
; 2; 4 ; 7; 10; 14), |
7) f = (0; 2; 5; 8 ; 11; 13), |
|||
8) f = (0; 1; 4; 5; 10; 12 ; 14), |
9) f = (0; 2; 6 |
; 8; 10; 14), |
|||
10) f = (1;3; 5 ; 9; 12 ; 14), |
11) |
f = (2 ; 7; 9 ; 11; 13), |
|||
12) f = (2;4 ; 6; 9 ; 14), |
13) |
f = (2;5 ; 10; 12 ; |
14), |
||
14) f = (0 |
; 3; 8; 9; 11; 12), |
15) f = (0; 1; 3 |
;5; 8; 10 |
;12; 14), |
|
16) f = (0 |
; 2; 5 ; 10; 14), |
17) |
f = (0;4; 6 |
; 8; 12;14), |
|
18) f = (3 |
; 8; 11; 12), |
19) |
f = (0 ; 4; 7;11; 12; 15). |
||
2. dLQ BULEWOJ FUNKCII g(a; b; c) SOSTAWXTE TABLICU ISTINNOSTI, NAJDITE MNOGO^LEN vEGALKINA, MINIMALXNU@ dnf I POSTROJTE KONTAKTNU@ SHEMU
0) g = ((a ! b) c) b, |
1) g = |
((a b) |
! |
|
c |
) b, |
||||||||||||||
|
|
|
||||||||||||||||||
2) g = ((b |
|
) _ |
|
) |
c, |
3) g = |
(a b)c |
|
|
, |
|
|
||||||||
a |
c |
a |
||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||
4) g = ( |
a |
(b _ c) abc) _ |
c ! a |
, |
5) g = a (b ! c) b, |
|||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
51 |
|
|
|
|
|
|
|
|
|
6) g = (a b) b ! c, 8) g = (a c) (b ! a),
10) |
g = (ab c) ! (b |
|
), |
||||||||||
c |
|||||||||||||
12) |
g = (a b |
|
) ( |
|
|
! b), |
|||||||
c |
a |
||||||||||||
14) |
|
|
|
|
|
|
|
|
|||||
g = (a c) (bc ! ac |
), |
||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
16) |
g = |
a (b |
! c) b, |
||||||||||
|
|
||||||||||||
18) |
g = ab ! c (c |
a |
), |
||||||||||
7) g = (a b |
(b ! c)) ! a |
c |
, |
|
|
|||||||||||||||||||||
9) |
|
= |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||
g |
|
(bc |
|
c a) ! |
||||||||||||||||||||||
(b _ c) |
! a, |
|
|
|
|
|
||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||
11) |
g = (a ! a bc) ! |
(ab b |
c |
), |
||||||||||||||||||||||
13) |
g = (a bc) ( |
b |
! |
|
|
|
|
), |
||||||||||||||||||
ac |
||||||||||||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
||||||||||
15) |
g = a ! (b c) b |
|
, |
|||||||||||||||||||||||
c |
||||||||||||||||||||||||||
17) |
g = (a |
|
|
bc) (a ! b), |
||||||||||||||||||||||
c |
||||||||||||||||||||||||||
19) |
g = b ( |
|
! (b |
|
)). |
|||||||||||||||||||||
ac |
a |
|||||||||||||||||||||||||
3.dOKAVITE SLEDU@]IE UTWERVDENIQ:
0)(A4B) n (A [ C) = (B \ C) n A;
1)(A [ C) n ( A4B) = (A n B) [ ((B \ C) n A);
2)(A4 B) [ (C n A) = (A \ B) [ A [ (B n C);
3)( B n A)4(A [ C) = (B4C) n A;
4)( A n C) [ (A4B) = (A n B) [ ((B [ C) n A);
5)(A4 B) n (A \ C) = ((A \ B) n C) [ ( B n A);
6)( A [ C) n (A4B) = ((A \ B) n C) [ A [ B ;
7)( A4B) \ (B n C) = A [ B [ C ;
8)(A n B)4(A \ C) = A \ (B4C);
9)( C n B) [ (A4 C) = (A \ C) [ C [ (A n B);
10)(A4B) n ( A \ C) = (A n B) [ ((B \ C) n A);
11)(A \ C) n (A4B) = (A \ B) n C ;
12)(A4B) \ ( B n C) = A n (B [ C);
13)( B n A)4( A \ C) = A [ (B4C);
14)A4(B n (A4C)) = (A n B) [ (B n C);
15)( A4B) n ( B [ C) = A \ B \ C ;
16)(A n C) \ (A4B) = (A \ C) n B ;
17)(A4 B) [ (B n C) = A [ B [ (B n (C n A));
18)(A n B)4( A [ C) = A n (B4C);
52
19) C4(B n (A4C)) = (C n B) [ (B n A).
4. dANY BINARNYE OTNO[ENIQ |
R1 |
A P |
I |
R2 |
|
B |
|
P , |
GDE |
|||||||||||||
|
|
|
|
|
|
|
|
|
|
1 |
|
|||||||||||
A = |
|
a; b; c |
|
, B = |
|
x; y; z |
|
, P |
= |
|
|
|
, |
|||||||||
f |
g |
f |
g |
f |
p; q; r; t |
g |
. nAJDITE |
|
R1 |
|
R; |
|||||||||||
ESLI |
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
0)R1 = f(a; p); (a; r); (b; q); (b; t);(c; r); (c; t)g, R2 = f(x; p); (y; p); (y; q); (z; q); (z; r); (z; t)g;
1)R1 = f(a; p); (a; q); (a; r); (a; t); (b; r); (c; t)g, R2 = f(x; p); (y; p); (y; q); (y; t); (z; q); (z; t)g;
2)R1 = f(a; p); (b; q); (b; r); (b; t);(c; r);(c; t)g, R2 = f(x; p); (x; q);(y; q);(y; r);(z; q);(z; r)g;
3)R1 = f(a; p); (a; q); (b; p); (b; q); (c; r); (c; t)g, R2 = f(x; q); (x; r); (y; q); (y; r); (y; t); (z; t)g;
4)R1 = f(a; p); (a; q); (a; r); (a; t); (b; r); (c; r)g, R2 = f(x; p); (x; q);(y; p); (y; r); (z; q); (z; r)g;
5)R1 = f(a; p); (a; q); (a; r); (b; p); (c; r); (c; t)g, R2 = f(x; p); (x; q);(y; q);(y; r);(z; p); (z; r)g;
6)R1 = f(a; p); (a; r); (b; r); (b; t);(c; r); (c; t)g, R2 = f(x; r); (y; q); (y; r); (y; t);(z; r); (z; t)g;
7)R1 = f(a; p); (a; q); (b; q); (c; q); (c; r); (c; t)g, R2 = f(x; p); (x; q);(y; q);(y; r);(z; q);(z; r)g;
8)R1 = f(a; p); (a; q); (b; q); (b; r); (c; q); (c; t)g, R2 = f(x; r); (y; p); (y; q); (y; r); (y; t); (z; r)g;
9)R1 = f(a; p); (b; p); (b; q); (b; r); (c; q); (c; r)g, R2 = f(x; p); (x; q);(x; r); (y; q); (y; r); (z; t)g;
10)R1 = f(a; p); (a; q); (b; q); (b; t); (c; p); (c; t)g, R2 = f(x; p); (x; q);(y; q);(y; r);(y; t); (z; p)g;
11)R1 = f(a; p); (a; q); (a; r); (b; q); (c; q); (c; r)g, R2 = f(x; p); (x; r);(y; q);(y; r);(z; q); (z; t)g;
12)R1 = f(a; p); (a; q); (b; q); (b; r); (c; r); (c; t)g, R2 = f(x; p); (x; r);(y; q);(y; r);(y; t); (z; r)g;
13)R1 = f(a; p); (a; q); (a; r); (b; q); (c; q); (c; r)g,
53
R2 = f(x; q); (x; r); (y; q); (y; r); (z; p); (z; t)g;
14)R1 = f(a; p); (a; r); (b; q); (b; r); (c; r); (c; t)g, R2 = f(x; r); (y; q); (y; r); (y; t);(z; r); (z; t)g;
15)R1 = f(a; p); (a; q); (a; r); (b; r); (b; t); (c; t)g, R2 = f(x; p); (x; t); (y; q); (y; r); (z; q); (z; r)g;
16)R1 = f(a; p); (a; q); (b; p); (b; q); (c; q); (c; t)g, R2 = f(x; r); (y; p); (y; q); (y; r); (y; t); (z; r)g;
17)R1 = f(a; p); (a; q); (a; r); (b; t);(c; q);(c; r)g, R2 = f(x; p); (x; q);(y; q);(y; r); (z; q); (z; t)g;
18)R1 = f(a; p); (b; p); (b; q); (b; r); (b; t); (c; t)g, R2 = f(x; p); (x; q);(y; r);(z; q); (z; r); (z; t)g;
19)R1 = f(a; q);(b; p); (b; q); (b; t); (c; r); (c; t)g, R2 = f(x; p); (y; q); (z; p); (z; q); (z; r); (z; t)g.
5. wY^ISLITE ZNA^ENIE WYRAVENIQ
0) |
2P3 + A42 ; 2C53; |
1) |
P5 ; 9A(3)2 ; 6C42; |
||
2) |
6P2 + A53 ; 3C(4)2 ; |
3) |
P (3; 0; 2) + A52 ; 2C64; |
||
4) |
P (2; 2; 1) + 2A43 ; 3C(3)5 ; |
5) |
P4 + A(2)4 ; C(5)3 ; |
|
|
6) |
P (3; 1; 2) + A(2)3 ; 6C52; |
7) |
2P (2; 1; 1) + A(3)4 |
; 5C(4)3 ; |
|
8) |
P2 ; A53 + 9C42; |
9) |
2P3 + A52 ; 2C(4)2 ; |
||
10) |
P4 ; A(3)2 ; C64; |
11) |
P5 ; A(3)4 ; C(3)5 ; |
|
|
12) |
P (3; 0; 2) |
; A43 + C52; |
13) |
P(2; 2; 1) + A42 ; C(5)3 ; |
|
14) |
P (3; 1; 2) |
+ A(2)4 ; 7C53; |
15) |
2P (2; 1; 1) + A(2)3 |
; C(4)3 ; |
16) |
P6 ; 7A64 ; 3C42; |
17) |
P6 ; A(5)4 ; C64; |
|
|
18) |
P (3; 0; 3) |
; 6A53 + C(5)3 ; |
19) |
2P (2; 0; 3) ; A(5)3 |
+ 3C(7)2 : |
6.rE[ITE UKAZANNYE REKURRENTNOSTI
0)a0 = 1, b0 = 2, an+1 = 3an + 2bn , bn+1 = an + 2bn ;
1)a0 = 5, a1 = ;1, an+2 = 2an+1 + 8an ; 9;
2)a0 = 4, b0 = ;1, an+1 = ;2an + 3bn , bn+1 = 2an ; bn ;
54
3)a0 = 5, a1 = 0, an+2 = an+1 + 12an ; 24;
4)a0 = 5, b0 = 0, an+1 = 2an ; 2bn , bn+1 = ;2an ; bn ;
5)a0 = 2, a1 = ;2, an+2 = 4an+1 + 5an + 24;
6)a0 = 3, b0 = ;4, an+1 = an + 2bn , bn+1 = 5an ; 2bn ;
7)a0 = 4, a1 = 0, an+2 = ;an+1 + 12an + 10;
8)a0 = 6, b0 = 3, an+1 = 3an ; 4bn , bn+1 = an ; 2bn ;
9)a0 = 6, a1 = 16, an+2 = 6an+1 ; 8an ; 6;
10)a0 = 0, b0 = 3, an+1 = an + 2bn , bn+1 = 4an ; bn ;
11)a0 = 2, a1 = 6, an+2 = 6an+1 ; 9an + 12;
12)a0 = 1, b0 = ;1, an+1 = 5an ; 3bn , bn+1 = 3an ; bn ;
13)a0 = 0, a1 = 6, an+2 = 8an+1 ; 16an + 18;
14)a0 = 1, b0 = ;1, an+1 = ;3an + bn , bn+1 = ;an ; bn ;
15)a0 = ;2, a1 = ;4, an+2 = 10an+1 ; 25an + 16;
16)a0 = ;1, b0 = 1, an+1 = an + 4bn , bn+1 = ;an + 5bn ;
17)a0 = ;1, a1 = ;18, an+2 + 4an+1 + 4an + 18 = 0;
18)a0 = 1, b0 = 2, an+1 = ;5an ; 2bn , bn+1 = 2an ; bn ;
19)a0 = ;4, a1 = ;4, an+2 + 6an+1 + 9an + 16 = 0.
7.nAJDITE MATRICU SMEVNOSTI DLQ GRAFA G(X; ;), ESLI MNOVES- TWO WER[IN X = fx1; x2; x3 ; x4g, A NABOR REBER ; UKAZAN NIVE:
0); = ((x1; x2)?3; (x1; x3); (x1; x4)?2; (x2; x3)?2; (x3; x4); (x3; x3));
1); = ((x1; x2)?2; (x1; x3); (x1; x4)?3; (x2; x3); (x3; x4)?2; (x1; x1));
2); = ((x1; x2); (x1; x3); (x1; x4)?2; (x2 ; x3)?2; (x3; x4)?3; (x4; x4));
3); = ((x1; x2)?2; (x1; x3); (x1; x4); (x2 ; x3)?3; (x3; x4)?2; (x4; x4));
4); = ((x1; x2)?3; (x1; x3); (x1; x4)?2; (x2; x3); (x3; x4)?2; (x3; x3));
5); = ((x1; x2); (x1; x4)?3; (x2; x4); (x2 ; x3)?2; (x3; x4)?2; (x1; x1));
6); = ((x1; x2)?2; (x1; x4)?2; (x2 ; x3); (x2; x4); (x3; x4)?3; (x4; x4));
7); = ((x1; x2); (x1; x3); (x1; x4)?2; (x2 ; x3)?3; (x3; x4)?2; (x4; x4));
8); = ((x1; x2)?3; (x1; x4)?2; (x2 ; x3); (x2; x4)?2; (x3; x4); (x3; x3));
55
9); = ((x1; x2); (x1; x3)?2; (x1; x4)?3; (x2; x3); (x3; x4)?2; (x3; x3));
10); = ((x1; x2); (x1; x4); (x2 ; x3)?2; (x2; x4)?2; (x3; x4)?3; (x3; x3));
11); = ((x1; x2); (x1; x4); (x2 ; x3)?3; (x2; x4)?2; (x3; x4)?2; (x4; x4));
12); = ((x1; x2)?3; (x1; x4)?2; (x2; x3); (x2; x4); (x3; x4)?2; (x3; x3));
13); = ((x1; x2)?2; (x1; x4)?3; (x2; x3)?2; (x2; x4); (x3; x4); (x4; x4));
14); = ((x1; x2)?2; (x1; x4); (x2; x3)?2; (x2; x4); (x3; x4)?3; (x3; x3));
15); = ((x1; x2); (x1; x4)?2; (x2; x3)?3; (x2; x4); (x3; x4)?2; (x4; x4));
16); = ((x1; x2) ? 3; (x1; x3); (x1; x4); (x2; x3); (x3; x4) ? 2; (x4; x4));
17); = ((x1; x2) ? 2; (x1; x4) ? 3; (x2; x3); (x2; x4); (x3; x4); (x4; x4));
18); = ((x1; x2); (x1; x3); (x1; x4); (x2; x3) ? 2; (x3; x4) ? 3; (x3; x3));
19); = ((x1; x2) ? 2; (x1 ; x4) ? 3; (x2; x3) ? 3; (x3; x4); (x4; x4)):
8.pOSTROJTE GRAF O2 + Gi ILI K2 + Gi , GDE GRAFY O2 , K2 I Gi IZOBRAVENY NA RIS. 28, I NAJDITE CIKLOMATI^ESKOE I HROMA- TI^ESKOE ^ISLO POLU^IW[EGOSQ GRAFA:
0) O2 + G1 , |
|
1) K2 + G1 , |
2) O2 + G6 , |
|
3) K2 + G6 , |
|
|||||||||||||
4) O2 + G2 , |
|
5) K2 + G2 , |
6) O2 + G7 , |
|
7) K2 + G7 , |
|
|||||||||||||
8) |
O2 + G3 , |
|
9) K2 + G3 , |
10) O2 + G8 , |
|
11) K2 + G8 , |
|
||||||||||||
12) |
O2 + G4 , |
|
13) |
K2 + G4 , |
14) |
|
|
O2 |
+ G9 , |
|
15) K2 + G9 , |
|
|||||||
16) |
O2 + G5 , |
|
17) |
K2 + G5 , |
18) |
|
|
O2 |
+ G10 , |
19) K2 + G10 . |
|
||||||||
|
b |
b |
b |
b |
|
|
b |
|
|
b |
|
|
b |
b |
b |
|
b |
b |
|
|
b |
|
|
|
|
||||||||||||||
|
G1b |
b G2b |
b |
|
|
b |
|
|
|
|
b |
|
b |
|
|
b |
|||
|
G3b |
|
|
G4b |
|
G5b |
|
G6b |
|||||||||||
|
b |
b |
|
b |
b |
|
b |
b |
b |
|
|
|
|
|
|||||
|
b |
|
b |
b |
|
|
b |
||||||||||||
|
|
|
b |
||||||||||||||||
|
G7b |
b G8b |
b G9b |
|
b G10b |
b |
|
||||||||||||
|
|
O2 |
|
|
K2 |
|
|||||||||||||
rIS. 28
56