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

Let, A, B and C be the sets such that  and.

To show: B = C

Let x ∈ B

$\Rightarrow x \in A \cup B \quad[B \subset A \cup B]$

$\Rightarrow x \in \mathrm{A} \cup \mathrm{C} \quad[\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cup \mathrm{C}]$

$\Rightarrow x \in \mathrm{A}$ or $x \in \mathrm{C}$

Case I

x ∈ A

Also, x ∈ B

$\therefore x \in A \cap B$

$\Rightarrow x \in \mathrm{A} \cap \mathrm{C} \quad[\because \mathrm{A} \cap \mathrm{B}=\mathrm{A} \cap \mathrm{C}]$

$\therefore x \in \mathrm{A}$ and $x \in \mathrm{C}$

$\therefore x \in \mathrm{C}$

$\therefore B \subset C$

Similarly, we can show that $\mathrm{C} \subset \mathrm{B}$.

$\therefore \mathrm{B}=\mathrm{C}$