Question

If $AandB$ are subsets of a set $X$, then what is $\left[A\cap (X-B)\right]\cup B$ equal to?

• A. $A\cup B$
• B. $A\cap B$
• C. $A$
• D. $B$

Answer 1

The correct option is A

$A\cup B$

Finding the value of $\mathbf{[}\mathit{A}\mathbf{\cap }\mathbf{(}\mathit{X}\mathbf{-}\mathit{B}\mathbf{]}\mathbf{\cup }\mathit{B}$:

Given that $AandB$ are subsets of a set $X$.

$\left[A\cap (X-B)\right]\cup B$ is the union of $AandB$

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

$\therefore \mathbf{A}\mathbf{\cap }\mathbf{(}\mathbf{X}\mathbf{-}\mathbf{B}\mathbf{)}\mathbf{\cup }\mathit{B}\mathbf{}\mathbf{=}\mathbf{}\mathit{A}\mathbf{\cup }\mathit{B}$

Hence, option A is the correct answer.