Question

If $A,B$ and $C$ are any $3$ sets then $A\times (B\cup C)$ is equal to

• A. $(A\times B)\cup (A\times C)$
• B. $(A\cup B)\times (A\cup C)$
• C. $(A\times B)\cap (A\times C)$
• D. None of these

Answer 1

The correct option is A

$(A\times B)\cup (A\times C)$

Explanation for the correct option:

Find the value of $A\times (B\cup C)$:

$\begin{array}{rcl}A\times (B\cup C)& =& A\times \left[\left(B\cup C\right)-\left(B\cap C\right)\right]\\ & =& A\times \left(B\cup C\right)-A\times \left(B\cap C\right)\\ & =& \left(A\times B\right)\cup \left(A\times C\right)-\left(A\times B\right)\cap \left(A\times C\right)\end{array}$

By using Distributive law, we get

$\mathit{A}\mathbf{}\mathbf{\times }\mathbf{}\left(B\cup C\right)\mathbf{}\mathbf{=}\mathbf{}\left(A\times B\right)\mathbf{}\mathbf{\cup }\mathbf{}\left(A\times C\right)$

Hence, option ‘A’ is Correct.