Question

Let A, B and C be the sets such that $A ∪ B = A ∪ C$ and $A ∩ B = A ∩ C$. show that B = C.

Answer 1

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