Contradictory definition in set theory book?

Question

I'm using a book that defines $A\setminus B$ (apparently this is also written as $A-B$) as $\{x\mid x\in A,x\not\in B\}$, but then there was an exercise that asked to find $A\setminus A$. Wouldn't it be a contradiction according to the definition given? (All $x$ such that $x$ is in $A$ and not in $A$...)

Note: The back of the book gives the answer $\emptyset$ with no explanation. Note2: The book is Introductory Mathematics by Geoff Smith.

Thanks.

Answer 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.

Answer 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.

Answer 3

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

Answer 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.

Answer 5

By the definition of set builder notation,

$$y \in \{ x \in A \mid x \notin A \} $$

if and only if

$$ y \in A \wedge y \notin A $$

Since the latter is a contradiction, we've proven that, for all $y$,

$$ y \notin \{ x \in A \mid x \notin A \} $$

However, we already know a set that tabulates that membership relation: i.e. we have

$$y \notin \varnothing$$

In particular, for all $y$, we have

$$ y \in \{ x \in A \mid x \notin A \} $$

if and only if

$$ y \in \varnothing $$

and so we conclude

$$ \varnothing = \{ x \in A \mid x \notin A \} $$