Page 87 - ICSE Math 7

Operations on Sets

Union of sets

Let A and B be any two sets. The union of sets A and B is the set Maths Info
of all the elements which belong to either A or B or both. Elements
common to both the sets are taken only once in the union set. It is • For any two sets A and B,
denoted by A ∪ B (read as A union B). A ∪ B = B ∪ A.
• For any three sets A, B and C,
For example, if A = {2, 3, 4, 5} and B = {3, 5, 7, 9, 11}, then A ∪ (B ∪ C) = (A ∪ B) ∪ C.
A ∪ B = {2, 3, 4, 5, 7, 9, 11}.

Elements 3 and 5 are in both A and B, but we write them only once as elements in a set cannot be repeated.

Example 8: Let A = {4, 6, 8, 10} and B = {3, 7, 9}. Find A ∪ B.
Solution: A ∪ B = {4, 6, 8, 10} ∪ {3, 7, 9} = {4, 6, 8, 10, 3, 7, 9}

Example 9: Let A = {a, b, c, d} and B = {c, f, g}. Verify that A ∪ B = B ∪ A
Solution: A ∪ B = {a, b, c, d} ∪ {c, f, g} = {a, b, c, d, f, g}
and B ∪ A = {c, f, g} ∪ {a, b, c, d}

= {c, f, g, a, b, d} = {a, b, c, d, f, g}
\ A ∪ B = B ∪ A

Intersection of sets
Let A and B be any two sets. The intersection of sets A and B is defined as the set which contains the
elements common to both A and B. It is denoted by A ∩ B (read as A intersection B).
For example, let A = {a, e, i, o, u} and B = {c, d, e, f, i}, then A ∩ B = {e, i}. Here elements e and i
belong to both the sets A and B.

Example 10: Let P = {x : x is an even number between 3 and 15}, and Q be the set of first six multiples
of 3. Find P ∩ Q.
Solution: Sets P and Q in roster form are given by:
P = {4, 6, 8, 10, 12, 14}, Q = {3, 6, 9, 12, 15, 18}

The common elements of P and Q are 6 and 12.
\ P ∩ Q = {6, 12}

Example 11: Let A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12, 15}. Find A ∩ B.
Solution: A ∩ B = {2, 4, 6, 8, 10, 12} ∩ {3, 6, 9, 12, 15} = {6, 12}

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.

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