OPERATION ON SETS –
(1) Union Of Sets –
The union of two sets A & B is a set C formed by combining the elements of A & B. It contains all elements in either of the sets A or B. Elements that are common to both A and B need to be listed only once in set C. The symbol A ∪ B means the union of A and B, and is read “A union B”.
A ∪ B = {x ǀ x ∈ or x ∈ B or x ∈ both A & B}
Example.1) If A = {1, 2, 3}, and B = {4, 5, 6}, A ∪ B = ?
Ans.) A ∪ B = {1, 2, 3, 4, 5, 6}
Example.2) If A = {p, q, r}, B = {r, s, t, x, y, z}, then find A ∪ B
Ans.) A ∪ B = {p, q, r, s, t, x, y, z}
Example.3) If A = {factor of 12}, and B = {factors of 16} then find A ∪ B
Ans.) A = {factor of 12} = {1, 2, 3, 4, 6, 12},
B = {factors of 16} = {1, 2, 4, 8, 16},
and A ∪ B = {1, 2, 3, 4, 6, 8, 12, 16}
Example.4) If A = {2, 3, 5, 7, 11}, B = {1, 3, 5, 7, 9}, C = {0, 1, 2, 3} then prove (A ∪ B) ∪ C = A ∪ (B ∪ C)
Ans.) A ∪ B = {1, 2, 3, 5, 7, 9, 11}
(A ∪ B) ∪ C = {1, 2, 3, 5, 7, 9, 11} ∪ {0, 1, 2, 3}
= {0, 1, 2, 3, 5, 7, 9, 11}
On the other hand,
B ∪ C = {1, 3, 5, 7, 9} ∪ {0, 1, 2, 3} = {0, 1, 2, 3, 5, 7, 9}
A ∪ (B ∪ C) = {2, 3, 5, 7, 11} ∪ {0, 1, 2, 3, 5, 7, 9}
= {0, 1, 2, 3, 5, 7, 9, 11}
So, we can say that, (A ∪ B) ∪ C = A ∪ (B ∪ C) (Proved)
Example.5) Is it true that for any sets A & B, P(A) ∪ P(B ) = P(A ∪ B) ? Justify your answer
Ans.) To prove, let us take an example suppose, A = {p}, B = {q}.
Then, A ∪ B = {p, q}
So, P(A) = Set of subsets of A = {ϕ, {p}}
P(B) = Set of Subsets of B = {ϕ, {q}}
So, P(A) ∪ P(B) = {ϕ, {p}, {q}} ……………………..(i)
A ∪ B = {p, q}
=> P(A ∪ B) = set of subsets of P(A ∪ B)
So, P(A ∪ B) = {ϕ, {p}, {q}, {p, q}} ……………………..(ii)
From (i) & (ii)is clear that P(A) ∪ P(B) ≠ P(A ∪ B)
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