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Question
For any two sets A and B, prove that A ⊂ B ⇒ A ∩ B = A
Solution
Let A ⊂ B. We need to prove A ∩ B = A.
For all x ∈ A
⇒ x ∈ A and x ∈ B (A ⊂ B)
⇒ x ∈ A ∩ B
⇒ A ⊂ A ∩ B
Also, A ∩ B ⊂ A
Thus, A ⊂ A ∩ B and A ∩ B ⊂ A
⇒ A ∩ B = A [Proved in (ii)]
∴ A ⊂ B ⇒ A ∩ B = A
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