# Solving Mathematical Proof

angela107
Homework Statement:
Is it TRUE that for all sets ##A## and ##B## the identity ##A \setminus (A \setminus B) =A ∩ B## holds?
Relevant Equations:
n/a
##A ∖ B## can't include any elements that are not in ##A##, so it is the same as saying ##A∖(A∩B)##; it's exactly the elements of ##A## except those in ##A∩B##.

##A∖(A∖(A∩B))## is exactly the elements of ##A## except those in (exactly the elements of ##A## except those in ##A∩B##). This is the same as ##A∩B##.

Therefore, it is true that for all sets A and B the identity ##A ∖ (A ∖B) =A ∩ B##holds.

Is this correct?

Well, I think you are going in the right direction. But a real formal proof (at this level) requires more details. Typically, when showing that two sets ##X,Y## are equal, you show that ##X \subseteq Y## and ##Y\subseteq X##. Showing ##X\subseteq Y## can be done by fixing an arbitrary element ##x\in X## and then after some steps deducing that ##x \in Y##. Similarly, you show ##Y \subseteq X##. So, let us try this on your case:

Let ##x \in A\setminus (A \setminus B)##. Then ##x\in A## and ##x \notin A \setminus B##. The latter means that ##x\notin A## or that ##x\in B##, but we already know that ##x\in A## so we must have ##x\in B##. Hence, ##x\in A## and ##x\in B##, which means ##x\in A \cap B##.

Can you try the other direction yourself now?

(1) Try to write a more descriptive title for your question. For example, "Prove the set equality ##A\cap B = A \setminus (A \setminus B)##"

PeroK
Homework Helper
Gold Member
2022 Award
In this case it may help to note that both sets are subsets of ##A##. You could then look at the two cases where ##x## is or is not in ##B##.

The moral for these questions is to stay calm and think logically!