For any two sets A and B, prove that A ∪ B = A ∩ B = A = B ⟺ A =

1.	For any two sets A and B, prove that A ∪ B = A ∩ B = A = B ⟺ A = B
Answer» Let A = B, then A ∪ B = A and A ∩ B = A A ∪ B = A ∩ B Thus, A = B …(i) Conversely, let A ∪ B = A ∩ B Now, let x ∈ A x ∈ (A ∪ B ) [∴ A ∪ B = A ∩ B] x ∈ (A ∩ B ) (x ∈ A and x ∈ B) x ∈ B A ⊆ B …(ii) Now, let y ∈ A y ∈ A ∪ B y ∈ A ∩ B[∴ A ∪ B = A ∩ B] y ∈ A and y ∈ B y ∈ A ∴ B ⊆ A …(iii) From equations (ii) and (iii), we get A = B

For any two sets A and B, prove that A ∪ B = A ∩ B = A = B ⟺ A = B

Answer»

Let

A = B, then A ∪ B = A and A ∩ B = A

A ∪ B = A ∩ B

Thus, A = B …(i)

Conversely, let

A ∪ B = A ∩ B

Now, let

x ∈ A

x ∈ (A ∪ B ) [∴ A ∪ B = A ∩ B]

x ∈ (A ∩ B )

(x ∈ A and x ∈ B)

x ∈ B

A ⊆ B …(ii)

Now, let

y ∈ A

y ∈ A ∪ B

y ∈ A ∩ B[∴ A ∪ B = A ∩ B]

y ∈ A and y ∈ B

y ∈ A

∴ B ⊆ A …(iii)

From equations (ii) and (iii), we get A = B