(a) Associativity: (
)
(
)
(
)
(
)
A B
C A
B C
A B
C A
B C
∪
∪ = ∪
∪
∩
∩ = ∩
∩
(b) Distributivity: (
)
(
) (
)
(
)
(
) (
)
A B
C
A C
B C
A B
C
A C
B C
∪
∩ =
∩
∪
∩
∩
∪ =
∪
∩
∪
16. Define the set operation viz., Absorption.
(
)
(
)
A B
A A
A B
A A
∪
∩ =
∩
∪ =
17. State DeMorgan’s Laws.
A B C
A B
A C
A B C
A B
A C
−
∪
=
−
∩
−
−
∩
=
−
∪
−
(
) (
) (
)
(
) (
) (
)
18. What are disjoint sets?
If A and B have no common elements i.e., A B
∩ = φ, then the sets A
and B are said to be disjoint.
19. Define cardinality of a set.
The Cardinality of a set A, written |A|, is the number of elements in
set A.
20. Define powerset.
The set of all subsets of A, written 2
A , is called The power set of set
A.
21. If a set has ‘n’ elements, how many elements does the powerset have?
The Powerset has 2
n elements.
22. Define Cartesian Product.
The set of all ordered pairs (x, y) where x A
∈ and y B
∈ is called
Cartesian product of the sets A and B, denoted by A B
× , i.e.,
A B
x y x A
y B
× =
∈
∈
{( , ) :
}
and
23. Define a relation.
A relation on sets S and T is a set of ordered pairs (s, t), whose
(a) s S
∈ (s is a member of S)
(b) t T
∈
(c) S and T need not be different
(d) The set of all first elements is the “domain” of the relation, and
(e) The set of all the second elements is the “range” of the relation.
24. What is an equivalence relation?
A subset R of A A
× is called an equivalence relation on A if R
satisfies the following conditions:
(i) ( , )
a a R
∈ for all a A
∈ (R is reflexive)
(ii) If ( , )
,
a b R
∈ then ( , )
b a R
∈ , then ( , )
a b R
∈
( R is symmetric)
(iii) If ( , )
a b R
∈ and ( , )
b c R
∈ , then ( , )
a c R
∈ (R is transitive)
Introduction
53