Chapter 6.2, Problem 22ES

To determine

Find the mistake in the following “proof” that for all sets $A\text{ and }B$, $\left(A-B\right)\cup \left(A\cap B\right)\subseteq A$.

Proof; Suppose $A\text{ and }B$ are sets, and suppose $x\in \left(A-B\right)\cup \left(A\cap B\right)$. If $x\in A\text{ then }x\in A-B$.

Then, by definition of difference, $x\in A\text{ and }x\in B$. Hence $x\in A$, and so $\left(A-B\right)\cup \left(A\cap B\right)\subseteq A$ by definition of subset.