(e) A B B A
− = − .
19. Let A
i
i = {
}
1, 2, 3, KK for i = 1, 2, 3, KK . Find (a) A i
i
n
=1
U , (b) A i
i
n
=1
I
20. Using Membership table show that
A
B C
A B
A C
∩
∪
=
∩
∪
∩
(
) (
) (
).
21. Using Venn diagram show that
A
B C
A B
A C
∪
∩
=
∪
∩
∪
(
) (
) (
)
22. Using set builder notation and logical equivalences show that
A B A B
∩ = ∪
23. State and prove De Morgan’s laws.
24. Prove by (a) Venn Diagram (b) Membership table:
(i) Commutative law (ii) Distoibutive law.
25. Given
{ }
A
a B
ab
= ∈
=
, ,
{ }, find A
2 , B
3 and AB.
26. Given
{ }
A
a B
ab
= ∈
=
, ,
{ } determine A
* , B
* and B
+
27. Given A and B are subsets of Σ
* and ∈∉ A, show that the equation
X AX B
=
∪ has a unique solution X A B
=
* .
28. Define Σ
+ in terms of Σ
* .
29. Given L
ab bc ca L
aa ac cb
1
2
=
=
{ , , },
{ , , } determine
(a) L
L
1
2
∪ (b) L
L
1
2
∩ (c) L L
1
2
⋅ (d) L L
1 2 .
30. What do you mean by the Kleene closure of set A?
31. What do you mean by ∈-free closure of set A?
32. Given A a aa B
a C
aa
=
=
=
{ , },
{ },
{ } show that
A B C
AB AC
(
)
∩
⊂
∩
.
33. A survey was conducted among 1000 people. Of these 595 are
democrats. 595 wear glasses and 550 like icecream. 395 of them are
democrats who wear glasses, 350 of them are democrats who like
icecream and 400 of them wear glasses and like icecreams; 250 of them
are democrats who wear glasses and like icecream.
(a) How many of them are not Democrats, who do not wear glasses,
and do not like icecreams?
(b) How many of them are Democrats, who do not wear glasses and do
not like icecreams?
34. It is known that at the “Catherine Assumption University”, 60 percent of
them play bridge, 70 percent jog, 20 percent play tennis and bridge, 30
percent play Tennis and jog, and 40 percent play bridge and jog. If
someone claimed that 20 percent of the Professors jog and play bridge
and Tennis, would you believe in this claim? Why?
Introduction
47