Page 80 - ICSE Math 8

Difference of two sets
The difference of the sets A and B is defined as the set of those elements which belong to A but not to B. It is
denoted by A – B (read as A minus B).
To find the difference of 2 sets (say A and B) follow the given steps.
Step 1: Find the elements common to both A and B.
Step 2: (a) To find A – B: Remove the common elements from the set A and write the remaining elements to
get A – B.
(b) To find B – A: Remove the common elements from the set B and write the remaining elements to
get B – A.
For example, let E = {3, 4, 5, 6, 7, 8} and F = {2, 4, 6, 8, 10, 12, 14}, then E – F = {3, 5, 7} and F – E =
{2, 10, 12, 14}.

As, E – F ≠ F – E, the difference of two sets is not commutative.

Example 12: Let A = {9, 11, 13, 17, 21} and B = {5, 7, 8, 9, 11, 15, 17}. Find the following sets.
(a) A – B (b) B – A
Solution: Common elements of A and B are 9, 11, 17.
(a) A – B = {13, 21} (Removing 9, 11, 17 from set A and writing the remaining
elements of set A)
(b) B – A = {5, 7, 8, 15} (Removing 9, 11, 17 from set B and writing the remaining
elements of set B)
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Example 13: If A = {x | x = 2n + 1, n ∈ N, n ≤ 4}, B = {x | x = n + 1, n ∈ N, n ≤ 5} and
C = {x | x is a factor of 20}, verify that:
(a) A – (B ∪ C) = (A – B) ∩ (A – C) (b) A – (B ∩ C) = (A – B) ∪ (A – C)
Solution: The sets in roster form are given by:
A = {3, 5, 7, 9}, B = {2, 5, 10, 17, 26}, C = {1, 2, 4, 5, 10, 20}
(a) B ∪ C = {2, 5, 10, 17, 26, 1, 4, 20}, A – B = {3, 7, 9}, A – C = {3, 7, 9}
A – (B ∪ C) = {3, 5, 7, 9} – {2, 5, 10, 17, 26, 1, 4, 20} = {3, 7, 9}
( A – B) ∩ (A – C) = {3, 7, 9} ∩ {3, 7, 9} = {3, 7, 9}
\ A – (B ∪ C) = (A – B) ∩ (A – C)
(b) B ∩ C = {2, 5, 10}, A – B = {3, 7, 9}, A – C = {3, 7, 9}
A – (B ∩ C) = {3, 5, 7, 9} – {2, 5, 10} = {3, 7, 9}
( A – B) ∪ (A – C) = {3, 7, 9} ∪ {3, 7, 9} = {3, 7, 9}
\ A – (B ∩ C) = (A – B) ∪ (A – C)
Complement of a Set

Let U be the universal set and A be any subset of U. The complement of set A is defined as the set of those
c
elements of U which are not in A. It is denoted by A′ or A . In other words, A′ = U – A.
For example, let U = {1, 2, 3, 4, 5, 6, 7} and A = {3, 5, 7}, then A′ = {1, 2, 4, 6}.

Points to remember
• n(A – B) = n(A) – n(A ∩ B) and n(B – A) = n(B) – n(A ∩ B)
• n(A′) = n(U) – n(A)
• n(A ∪ B)′ = n(U) – n(A ∪ B)
• n(A ∩ B)′ = n(U) – n(A ∩ B)
Example 14: Let U = {x : x is a student of class VIII} and A = {x : x is a girl of class VIII}. Find A′.
Solution: A′ = U – A = {x : x is a boy of class VIII}

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