Q.For all sets $A$ and $B$, $A - (A \cap B)$ is equal to ______________.

The key idea here is Set Operations — specifically, the set difference and intersection.

Step 1: Recall that $A - (A \cap B)$ means all elements in $A$ that are not in $A \cap B$. …

This is a classic set theory identity, and it’s best understood by thinking about what each operation actually does to the elements.

Set difference $X - Y$ means “everything in $X$ that is not in $Y$.”

Intersection $A \cap B$ means “everything that is in both $A$ and $B$.”

So the expression $A - (A \cap B)$ asks: Take all of $A$, and remove from it the elements that belong to both $A$ and $B$.

What’s left? Only those elements of $A$ that are not in $B$ — because any element of $A$ that is also in $B$ gets removed. That’s exactly the definition of $A - B$.

Let’s verify this step by step.

1. Start with the definition of set difference. For any element $x$,

2. Unpack the “not in intersection” condition.

   $x \notin (A \cap B)$ means it is not true that $x$ is in both $A$ and $B$. In other words, either $x \notin A$ or $x \notin B$.

   But we already know from step 1 that $x \in A$. So the “$x \notin A$” case is impossible. Therefore, we must have $x \notin B$.

3. Combine the conditions.

   From step 1 and step 2, we get:

That is precisely the condition for $x \in A - B$.

4. Conclusion of the logic. We have shown: $x \in A - (A \cap B) \iff x \in A - B.$ …