Suppose that A and B are sets. Prove that A ∩ (B \ A) is the empty set

Sure, here is a step-by-step proof:

1. Let's start by understanding what the sets A and B \ A represent. Set A is just a set of elements. B \ A represents the set of all elements that are in B but not in A.

2. Now, let's consider the intersection of A and (B \ A), denoted as A ∩ (B \ A). By definition, an element is in the intersection of two sets if and only if it is in both sets.

3. For an element to be in A ∩ (B \ A), it would have to be in both A and in (B \ A) at the same time.

4. However, by definition, if an element is in (B \ A), it cannot be in A. Similarly, if an element is in A, it cannot be in (B \ A).

5. Therefore, there are no elements that can be in both A and (B \ A) at the same time.

6. This means that the intersection of A and (B \ A), or A ∩ (B \ A), is the empty set.

So, we have proved that A ∩ (B \ A) is the empty set.