Contradictory definition in set theory book?

Solution 1:

There is no contradiction. The set of all $x$ such that $x\in A$ and $x\notin A$ is empty. Since it is impossible that $x$ is both in $A$ and not in $A$ simultaneously.

Solution 2:

The other answers are, of course, all correct. But here's another way to think about it, that will, perhaps, build on your intuition.

You're right that there is a contradiction. The statements "$x \in A$" and $x \notin A$ are mutually exclusive, or contradictory. Therefore, the statement that "$x \in A \mathrel{\mathrm{and}} x \notin A$" must be false for all values of $x$.

Consequently, the set definition is equivalent to $\{ x \mid \mathrm{False} \}$. Therefore, this set must be empty.

Solution 3:

There is no element that is in $A$ and not in $A$ at the same time, so the answer is the empty set.

Solution 4:

It is useful to note that $A - B = A \cap B^C$. Taking $B = A$, we see $A- A = A \cap A^C = \emptyset$ by definition of complement.