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Question
If A and B are subsets of the universal set U, then show that A ⊂ B ⇔ A ∪ B = B
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Solution
A and B are subsets
To prove: A ⊂ B ⇔ A ∪ B = B
Proof: Let x ∈ A ∪ B
⇒ x ∈ A or x ∈ B
Since, A ⊂ B, we get,
⇒ x ∈ B
⇒ A ⊂ B ⊂ B ......(i)
We know that,
B ⊂ A ∪ B ......(ii)
From equations (i) and (ii),
We get,
A ∪ B = B
Now, Let y ∈ A
⇒ y ∈ A ∪ B
Since, A ∪ B = B, we get,
⇒ y ∈ B
⇒ A ⊂ B
So, A ⊂ B ⇔ A ∪ B = B
Hence Proved.
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