If A and B are non-empty subsets of a set C, then A ∪ (A ∩ B) is equal to

The absorption law of set theory states that for any two sets A and B, uniting a set with its own intersection with another set returns the original set: A ∪ (A ∩ B) = A. This follows directly from the fact that an intersection A ∩ B is always a subset of A, so it can never add any element beyond what A already contains.

1. A ∩ B contains only the elements common to both A and B, so every element of A ∩ B is already an element of A; hence A ∩ B ⊆ A.

2. Taking the union of a set with any of its own subsets cannot add new elements — for any set X and any subset Y ⊆ X, X ∪ Y = X.

3. Applying this with X = A and Y = A ∩ B (since A ∩ B ⊆ A from the previous step) gives A ∪ (A ∩ B) = A.

This can be verified with a concrete example: let C = {1, 2, 3, 4, 5}, A = {1, 2, 3}, and B = {2, 3, 4}. Then A ∩ B = {2, 3}, and A ∪ (A ∩ B) = {1, 2, 3} ∪ {2, 3} = {1, 2, 3}, which is exactly A — confirming the identity holds.