Question

Show that if A and B are sets, then$A-\left(A-B\right)=A\cap B$.

Short answer

$A-\left(A-B\right)=A\cap B$

Step-by-step solution

Step: 1

$$\begin{gathered} A-\left(A-B\right)=A-\left(A\cap \overline{B}\right) \\ =A\cap \left(A\cap \overline{B}\right) \\ =A\cap \left(\overline{A}\cup B\right) \\ =\left(A\cap \overline{A}\right)\cup \left(A\cap B\right) \\ =\cancel{O}\cup \left(A\cap B\right) \\ =A\cap B \end{gathered}$$