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Question
Given A = {0, 1, 2, 4, 5}, B = {0, 2, 4, 6, 8} and C = {0, 3, 6, 9}. Show that A ∪ (B ∪ C) = (A ∪ B) ∪ C i.e. the union of sets is associative.
Solution
A = {0, 1, 2, 4, 5}
B = {0, 2, 4, 6, 8}
C = {0, 3, 6, 9}
B ∪ C = {0, 2, 4, 6, 8} ∪ {0, 3, 6, 9}
= {0, 2, 3, 4, 6, 8, 9}
∴ A ∪ (B ∪ C)
= {0, 1, 2, 4, 5} ∪ {0, 2, 3, 4, 6, 8, 9}
⇒ A ∪ (B ∪ C)
= {0, 1, 2, 3, 4, 5, 6, 8, 9} ......(i)
A ∪ B = {0, 1, 2, 4, 5} ∪ {0, 2, 4, 6, 8}
= {0, 1, 2, 4, 5, 6, 8}
∴ (A ∪ B) ∪ C
= {0, 1, 2, 4, 5, 6, 8} ∪ {0, 3, 6, 9}
⇒ (A ∪ B) ∪ C = {0, 1, 2, 3, 4, 5, 6, 8, 9} ...(ii)
From (i) and (ii), we get
A ∪ (B ∪ C) = (A ∪ B) ∪ C
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