Контрольная работа - Основы дискретной математики
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1. (AB) = AB, .
:
:
= {a| aA aB}
= {a| aA i aB}
=U-A={a| aU i aA}
(AB) (AB) –
(AB)) –
((A) (aB)) –
A) (aB) –
(A) (aB) –
() (B) –
. (AB) –
2. (AB) = AB, .
(AB) = AB
:
:
= {a| aA aB}
= {a| aA i aB}
=U-A={a| aU i aA}
(AB) (AB) –
(AB)) –
((A) (aB)) –
A) (aB) –
(A) (aB) –
() (B) –
. (AB) –
3. XOY, X,Y{,U,A,A},O {,,\,÷}.
:
XOY, X,Y{,U,A,A},O
{,,\,÷}:
( - ; U – ; A – ; A - )
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,,\,÷, :
= {a| aA aB}
= {a| aA i aB}
= {a| aB i aA}
={a| aB i aA aA i aB}
, :
:
A = A
A = A
:
= A
U = A
:
A= U
A=
4., .
) ( A\A) (÷A) (UA);
) (AA( AUA);
) BC( ABC)( ABC).
:
) ( A\A) (÷A) (UA)
: A\A = A; ÷A = A; UA = U
: AAU
AAU = () = A(AU) = ( 3) = AA = A.
: ( A\A) (÷A) (UA) =
) (AA( AUA)
A((A( UA))
A =A A(A( UA))
UA=A A(AA)
AA=A AA = U –
: (AA( AUA)= U
) BC( ABC)( ABC) =
= BC(BCA))=
= BC(BCU)= BC((BC)U)=
=B(C(BC))= B((CB)(CC))=
= B((CB)U)=BU= U
BC( ABC)( ABC) =
= BC((A)BC)=
= (BC) ((A)BC)=
=(BC)((A)BC)=
=(BC)((A)(BC))=
=(BC)(U(BC))=
3
=(BC)(BC)= U
: BC( ABC)( ABC) = U
5., 52
3n-2
+ 3
3n-1
19.
:
, b , ,
a=mb.
1) , 19 n=1.
:
113213
325
, n=1 19, 19.
2) , n=k,
19,
12
34215
3
3
4
25
325
3333
1323
kkkk
kk
19.
, n=k+1
12
31082120
39210325325
33
3323131)1(32)1(3
1
kk
kkkkkk
k
S
3
319
12
34215(8
12
376
12
34215(8
12
3763322120
3)333)33333 kkkkkkkkk
,
1)1(32)1(3
1
325
kk
k
S 19.
, , n=1,
n=k, n=k+1.
, 19 n.
6. – 5 {}.
?
:
8- {}.
, ,
, (8-1=7) .
3 : 8-2=6 .
4 : 8-3=5 .
5 : 8-4=4 .
, }
8*7*6*5*4= 6720 , , .
: 6720
7. – 5 {},
. ?
:
7- {}.
,
. (7-1=6)
.
, .
(7-1=6) .
(7-1=6) .
(7-1=6) .
4
, }
7*6*6*6*6= 9072 , , .
: 9072 .
8. 3 18 . 1
10 .
?
:
1 3 10 13 .
:
3
!
2
!
1
!3
1
3
C
:
43758
40320
1764322560
!
8
!
10
!18
10
18
C
: 131274437583
10
18
1
3
CC
: 131274
9. , „”?
:
„” :
– 2 (k
1
)
– 3 (k
2
)
– 2 (k
3
)
– 1 (k
4
)
– 1 (k
5
)
– 1 (k
6
)
– 10 (n)
!!!
!
),,,(
621
621
kkk
n
kkkC
n
151200
24
3628800
!
1
!
1
!
1
!
2
!
3
!
2
!10
10
C
: 151200
10. 4 ?
:
64 (n) 4 (k).
,
, .
, , 64 .
, (64-1=63) .
, 63 .
, (64-2=62) .
62 .
(64-3=61) . 61 .
,
(64*63*62*61=15 249 024) .
: 15 249 024 .
11. 14 ,
– 13, – 12, – 11, – 7, – 6.
5
,
. ?
:
, , 1
.
6 : 141312116=1009008
: 1009008 .
12.
1
2
3
4
5
=12?
:
1
,
2
,
3
,
4
,
5
, 12:
12
5
1
i
i
x
5- m, 1 – , 2 – ,
... 5 – .
, :
m
mn
C
1
, n=5 m=12
4368
5
4
3
2
1
1615141312
!
11
!
5
!16
5
16
12
1125
CC
: 4368 .
13. 5 ?
:
k =2, n = 5
1) 5 - 0.
5! = 1*2*3*4*5 = 120 .
0 - 5. 5! =
1*2*3*4*5 = 120 .
240 .
2) 4 - 1.
4! = 1*2*3*4 = 24 . – 5
. 24*5 = 120 .
1 , - 4. 4!*5
= (1*2*3*4)*5 = 120 .
240 .
3) 3 - 2.
.
2 :
)!(!
!
mnm
n
C
m
n
=>
10
)!25(!2
!5
2
5
C
2 2! = 1*2 = 2 ;
3 3!= 1*2*3=6 .
2!*3!*10=6*2*10=120 .
2 - 3.
.
2 :
)!(!
!
mnm
n
C
m
n
=>
10
)!35(!3
!5
3
C
3 3!= 1*2*3=6 .
6
2 2! = 1*2 = 2 ;
3!*2!*10=6*2*10=120 .
240 .
4) 5 (240+240+240 =720) .
: 720 .
14.
) {1,2,3}×{a,b},
) {x,y}×{x,y,z},
) ×{a,b}.
:
)
1
={1,2,3};
2
={a,b},
1
2
= {1,2,3}×{a,b}= {(1,), (1,b), (2,a), (2,b), (3,a), (3,b)}
)
1
={x,y},
2
={x,y,z},
1
2
={x,y}×{x,y,z}= {(x,x), (x,y), (x,z), (y,x), (y,y), (y,z)}
)
1
=,
2
={a,b}
1
2
=.×{a,b}=
15. R = {(x,y)| x+y=x
2
–y
2
}, {1,2,-
1,0,7,8,9}.
:
x y
x+y=x
2
–y
2
2 1
-1 1
1 -1
0 -1
1 0
0 0
8 7
9 8
: (2,1); (-1,1); (1,-1); (0,-1); (1,0); (0,0); (8,7); (9,8).
16. {(1,2),(2,3),(1,3)}.
:
R={(1,2),(2,3),(1,3)} . ,
.
()
:
R
={(1,2),(2,3),(1,3),(1,1),(2,2),(3,3)}
: R={(1,2),(2,3),(1,3),(1,1),(2,2),(3,3)}
17. {(1,2),(2,3),(1,3)}.
:
R={(1,2),(2,3),(1,3)} . ,
.
, :
R
s
={(1,2),(2,3),(1,3),(2,1),(3,2),(3,1)}
: R
s
={(1,2),(2,3),(1,3),(2,1),(3,2),(3,1)}
18. {(a,b),(d,c),(c,b),(b,c)}.
7
:
R= {(a,b),(d,c),(c,b),(b,c)} . ,
.
(a,b) (b,c), ()
(d,c) (c,b), (d,b)
(c,b) (b,c), ()
(b,c) (c,b), (b,b)
:
R
t
{(a,b),(d,c),(c,b),(b,c), (),(d,b),(),(b,b)}.
, .
:
R
t
{(a,b),(d,c),(c,b),(b,c), (),(d,b),(),(b,b)}.
: R
t
{(a,b),(d,c),(c,b),(b,c), (),(d,b),(),(b,b)}.