How would you prove that A \ (A \ B) = A ∩ B is true for all sets A and B?

Question

I know how to prove this using a picture of a venn diagram but I am getting stuck solving it using letters. This is what I've tried:

Let x $\in$ A $\setminus$(A$\setminus$B)

x $\in$ A such that x $\notin$ B

x $\in$ A such that x $\notin$ A

Therefore x $\in$ A $\cap$ B

I feel like there are some steps before the last one that I'm missing

Any help is appreciated

Answer 1

Your deduction seems a bit off. The formal way of doing it would be $$\begin{array}{rcl} A \setminus (A\setminus B) &=& \{x \in A \mid x \notin A \setminus B\}\\ &=& \{ x \in A \mid x \in B\} \\ &=& A \cap B \end{array}$$ Note that the second identity comes from the fact that we require the elements $x$ to be in $A$ but not not in $B$, so they have to lie in $B$.