Use a follow-the-element proof method to prove that if A and B aresets, then (A ∪ B)C ⊆ (AC ∩ BC )

Sure, here is a step-by-step proof using the follow-the-element method:

1. Assume x is an arbitrary element of (A ∪ B)C. This means that x is in the complement of the union of A and B.

2. By definition of set complement, x is not in the union of A and B. This means that x is not in A and x is not in B.

3. By definition of set intersection, x is in the intersection of AC and BC if and only if x is in AC and x is in BC.

4. Since x is not in A and x is not in B, x is in AC and x is in BC.

5. Therefore, x is in the intersection of AC and BC.

6. Since x was an arbitrary element of (A ∪ B)C, we can conclude that (A ∪ B)C ⊆ (AC ∩ BC).

This completes the proof.