Question 3:
Let A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C. show that B = C.
Answer
Let, A, B and C be the sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C.
To show: B = C
Let x ∈ B
⇒ x ∈ A ∪ B [B⊂ A ∪ B]
⇒ x ∈ A ∪ C [A ∪ B= A ∪ C]
⇒ x ∈ A or x ∈ C
Case I
x ∈ A
Also, x ∈ B
⇒ x ∈ A ∩ C [∵ A ∩ B = A ∩ C]
∴ x ∈ A and x ∈ C
∴ x ∈ C
∴ B ⊂ C
Similarly, we can show that C ⊂ B.
∴ B = C
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