ДКР1 (Дискретка)
.docВарианты домашней контрольной работы 1 по дисциплине
"Дискретная математика"
Задача 1. Используя диаграммы Венна, доказать равенства:
|
1.
(A(BC))
2.
3.
4.
5. A (BC)= (AB) (AC) 6. A\(BC) = (A\B)(A\C) 7. A\B = A\ (A B) 8. A(B\C) = (A B) \ (AC) 9. (A B) \ (AC)= (A B) \C 10. (A\B)\C= (A\C)\ (B\C) 11. AB = A (B\A) |
12. A (B \A)= 13.
(A B)(A 14.
(A B)
(A 15. (A B) \B =A\B 16.
(A B)
\ 17.
( 18.
( 19.
A= (A 20.
21.
B(AC)=
(A\ 22.
23.
(D\ |
=A\C.
.
)
=A
)
=A
=A
)B
=B\A
)A
= A \B
)
(A B)
)
(C B)
)
(C A)
=A(DC)
Задача 2. Используя алгебру множеств, доказать равенства:
|
1.
(A(BC))
2.
3.
4.
5. A (BC)= (AB) (AC) 6. A\(BC) = (A\B)(A\C) 7 A\ (A B) =A\B 8. A(B\C) = (A B) \ (AC) 9. (A B) \ (AC)= (A B) \C 10. (A\B)\C= (A\C)\ (B\C) 11. AB = A (B\A) |
12.
(A B)(A 13. A (B \A)= 14. (A B) \B =A\B 15.
(A B)
(A 16.
(A B)
\ 17.
( 18.
( 19.
A= (A 20.
21.
B(AC)=
(A\ 22.
23.
(D\ |
=A
B
.
)
=A
)
=A
=A
)B
=B\A
)A
= A \B
)
(A B)
)
(C B)
)
(C A)
=A(DC)
Задача 3. Используя таблицы истинности, доказать равенства:
|
1.
(A(BC))
2. (A B) \B =A 3.
4.
5. A (BC)= (AB) (AC) 6. A\(BC) = (A\B)(A\C) 7. A\B = A\ (A B) 8. A(B\C) = (A B) \ (AC) 9. (A B) \ (AC)= (A B) \C 10. (A\B)\C= (A\C)\ (B\C) 11. AB = A (B\A) 12. A (B \A)= |
13.
(A B)(A 14.
(A B)
(A 15.
16.
(A B)
\ 17.
( 18.
( 19.
A= (A 20.
21.
(D\ 22.
23.
B(AC)=
(A\ |
=A
B
)
=A
)
=A
=B
)B
=B\A
)A
= A \B
)
(A B)
)
(C A)
=A(DC)
)
(C B)
Задача 4. Заданы множества А и В. Необходимо:
1) описать различными способами (матричным, графическим и в виде графа) бинарное отношение PAB;
2) найти обратное отношение P–1;
3) задано отношение QCD. Найти орграф для Q и построить орграф композиции PQ.
Варианты.
1. A={a, b, c, d}, B={1, 2, 7, 10}. C={1, 2, 7, 10}, D={f, k, t, m}.
P={(a,1), (b,10), (a,2), (c,7)}, Q={(1,k), (2,m), (10,f), (7,m)}.
2. A={1, 2, 7, 10}, B={ a, b, c, d }, C={ a, b, c, d}, D={f, k, t, n }.
P={(1,a), (10, b), (2, a), (7, c)}, Q={(a, n), (b,k), (b,n), (d,t)}.
3. A={1, 4, 17, –1}, B={ 4, 6, 3, 1 }, C={ 4, 6, 3, 1 }, D={–2, 5, –5, 8}.
P={(1,6), (4, 4), (–1, 3), (17, 1)}, Q={(4, 5), (6,–2), (1,–5), (3,8),(3,5)}.
4. A={ a, b, c, d }, B={ 12, –5, 8, 0}, C={ 12, –5, 8, 0}, D={N, g, c,k}.
P={(a,12), (c, 0), (d, –5), (b, 12)}, Q={(12, g), (8, N), (–5, c), (0,g),(0,c)}.
5. A={1, 4, 17, –1}, B={ 4, 6, 3, 1 }, C={ 4, 6, 3, 1 }, D={f, k, n, q}.
P={(1, 4), (4, 6), (–1, 3), (17, 1)}, Q={(4, f), (6, k), (1,q), (3,n),(3,k)}.
6. A={ f, k, c, v }, B={1, 2, 7, 10}, C={1, 2, 7, 10}, D={ a, b, c, d }.
P={(f,1), (c, 10), (k, 7), (v, 2)}, Q={(7, a), (2, b), (10, d), (1, d)}.
7. A={1, 2, 7, 9}, B={ 4, 6, 3, 10 }, C={ 4, 6, 3, 10 }, D={8, 9, 0, 13}.
P={(1,6), (2, 4), (7, 3), (9, 10)}, Q={(4, 8), (6, 0), 10,8), (3, 13),(3,0)}.
8. A={ a, b, c, d, r}, B={ 4, 6, 3, 5, 10 }, C={ 4, 6, 3, 5, 10}, D={8, 9, 0, 8, –3}.
P={(a,4), (c, 10), (r, 6), (d, 3), (b, 5)}, Q={(6, 8), (3, 0), (10, –3), (5, 9), (4, 9)}.
9. A={1, 4, 0, 5}, B={ 4, 6, 2, 0 }, C={4, 6, 2, 0 }, D={f, k, t, m}.
P={(1,4), (0, 0), (5, 6), (4, 2), (0, 4)}, Q={(6,k), (2,m), (4,f), (0,m)}.
10. A={1, 2, 7, 10}, B={ ee, b, c, d }, C={ ee, b, c, d}, D={f, k, t, n }.
P={(1,ee), (10, b), (2, ee), (7, c)}, Q={(c, n), (ee,k), (b,n), (d,t)}.
11. A={–1, 4, 17, 1}, B={– 4, 6, 3, 1 }, C={ –4, 6, 3, 1 }, D={f, k, n, q}.
P={(–1, –4), (4, 6), (–1, 3), (17, 1)}, Q={(–4, f), (6, k), (1,q), (3,n),(3,k)}.
12. A={ f, k, c, v }, B={1, 10, 7, 22}, C={1, 10, 7, 22}, D={ a, b, c, dw }.
P={(f,1), (c, 10), (k, 7), (v, 22)}, Q={(7, a), (22, b), (10, d), (1, dw)}.
13. A={11, 4, 10, 5}, B={ 4, 6, 2, 0 }, C={4, 6, 2, 0 }, D={f, k, t, m}.
P={(11,4), (10, 0), (5, 6), (4, 2), (5, 4)}, Q={(4,k), (6,m), (0,f), (2,m)}.
14. A={ q, b, c, d }, B={ 12, –5, 8, 10}, C={ 12, –5, 8, 10}, D={N, g, c,k}.
P={(q,12), (c, 10), (d, –5), (b, 12)}, Q={(–5, g), (12, N), (8, c), (10,g),(10,c)}.
15. A={ u, b, c, d, r}, B={ 4, 6, 3, 15, 10 }, C={ 4, 6, 3, 15, 10 }, D={8, 9, 0, 8, –3}.
P={(u,4), (c, 10), (r, 6), (d, 3), (b, 15)}, Q={(6, 8), (3, 0), (10, –3), (4, 9), (15, 9)}.
16. A={3, 4, 17, –1}, B={ 4, 6, –3, 1 }, C={ 4, 6, –3, 1}, D={f, k, n, q}.
P={(3, 4), (4, 6), (–1, –3), (17, 1)}, Q={(4, f), (6, k), (1,q), (–3,n),(–3,k)}.
17. A={8, 2, 7, 10}, B={ e, b, c, d }, C={ e, b, c, d}, D={f, k, t, p }.
P={(8,e), (10, b), (2, e), (7, c)}, Q={(e, p), (b,k), (c,p), (d,t)}.
18. A={m, b, c, d}, B={1, 20, 7, 10}. C={1, 20, 7, 10}, D={f, k, t, m}.
P={(m,1), (b,10), (d,20), (c,7)}, Q={(1,k), (20,m), (10,f), (7,m)}.
19. A={1, 4, –4, 5}, B={ 4, 6, 2, 0 }, C={4, 6, 2, 0 }, D={f, k, t, m}.
P={(1,4), (–4, 0), (5, 6), (4, 2), (–4, 4)}, Q={(6,k), (4,m), (2,f), (0,m)}.
20. A={1, 4, 10, 5}, B={ 4, 6, 2, 0 }, C={4, 6, 2, 0}, D={f, kq, t, cm}.
P={(1,4), (10, 0), (5, 6), (4, 2), (0, 4)}, Q={(0,kq), (2,m), (4,f), (6,cm)}.
21. A={1, 2, 7, 10}, B={ e, b, g, d }, C={ e, b, g, d }, D={f, k, t, p }.
P={(1,e), (10, b), (2, e), (7, g)}, Q={(e, p), (b,k), (g,p), (d,t)}.
22. A={1, 4, 17, –1}, B={ 4, 6, 3, 1 }, C={ 4, 6, 3, 1 }, D={–2, 5, –5, 8}.
P={(1,6), (4, 4), (–1, 3), (17, 1)}, Q={(4, 5), (6,–2), (1,–5), (3,8),(3,5)}.
23. A={5, 4, 16, –1}, B={ 4, 6, –3, 1 }, C={ 4, 6, –3, 1 }, D={f, k, n, q}.
P={(5, 4), (4, 6), (–1, –3), (16, 1)}, Q={(4, f), (6, k), (1,q), (–3,n),(1,k)}.
Задача 5. Пусть заданы множества U, A, B, C. Необходимо найти для множеств A, B, C характеристические вектора a, b,c и на их основе найти следующие множества T, V, D. Проверить полученные результаты непосредственным вычислением множеств T, V, D.
Варианты.
1.
,
,
,
,
,
.
2.
,
,
,
,
,
.
3.
,
,
,
4.
,
,
.
5.
,
,
6.
,
,
.
7
,
,
.
8.
,
.
9.
,
,
.
10.
,
.
11.
,
.
12.
,
.
13.
,
.
14.
,
,
.
15.
,
,
.
16.
,
,
.
17.
,
,
V=D\T.
18.
,
,
.
19.
,
,
.
20.
,
,
21.
,
.
22.
,
,
23.,
,
,
.