To prove this equation, we need to show that the left-hand side (LHS) is equal to the right-hand side (RHS).

Let's start by expanding the LHS using the definition of set difference:

A \ (A ∩ B) = {x | x ∈ A and x ∉ (A ∩ B)}

Since x ∉ (A ∩ B) means x is not in the intersection of A and B, we can rewrite it as:

A \ (A ∩ B) = {x | x ∈ A and x ∉ B}

This means that the elements in A that are not in B are in the set A \ (A ∩ B).

Now, let's look at the RHS:

A ∩ B = {x | x ∈ A and x ∈ B}

This means that the elements in both A and B are in the set A ∩ B.

If we compare the definitions of the LHS and RHS, we can see that they are actually the same. The elements in A that are not in B are the same as the elements in both A and B.

Therefore, we have shown that A \ (A ∩ B) = A ∩ B.