TITLE: Prove that $B \setminus (B \setminus A) = A$
QUESTION [7 upvotes]: Prove that $B \setminus (B \setminus A) = A$

My conclusion is that  $B \setminus (B \setminus A) \neq A$, OR, to be precise, $B \setminus (B \setminus A) = A \cap B$.

If I'm wrong, please help me prove it. If I'm correct, how should I show that the question asked cannot be proved?

REPLY [2 votes]: A truth table can be used as a logical Venn diagram. Each disjoint area in a Venn diagram corresponds to the appropriate member functions.
Using the definition $x\in B\setminus A\equiv x\in B \land x\notin A$ gives the following truth table.




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




$0$
$0$
$0$
$0$
$0$


$0$
$1$
$1$
$0$
$0$


$1$
$0$
$0$
$0$
$0$


$1$
$1$
$0$
$1$
$1$




As the last two columns are identical, we can conclude that $B\setminus(B\setminus A)\equiv A\cap B$.
A counter-example to the original conjecture is any $A\ne\emptyset$ such that $A\cap B=\emptyset$.