Sets & Operations — Given U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, and C = {c, d, e}, compute (A ∩ B) ∪ (A ∩ C).

Introduction / Context:
This is a direct practice of intersection and union operations on finite sets. We compute two intersections with A and then take their union. Careful element-wise checking avoids mistakes.

Given Data / Assumptions:

• U = {a, b, c, d, e, f}
• A = {a, b, c}
• B = {c, d, e, f}
• C = {c, d, e}

Concept / Approach:
Intersection keeps only common elements; union combines elements without duplication. Compute A ∩ B and A ∩ C separately, then unite the results.

Step-by-Step Solution:
A ∩ B = { elements in both A and B } = {c}A ∩ C = { elements in both A and C } = {c}(A ∩ B) ∪ (A ∩ C) = {c} ∪ {c} = {c}

Verification / Alternative check:
Since c is the only element common to A with either B or C, any union of these intersections must be {c}.

Why Other Options Are Wrong:
{a} and {b} are not in both A and B (or A and C). {d} appears in B and C but not in A; {c, d} incorrectly adds d.

Common Pitfalls:
Confusing union with intersection; or accidentally including elements that are not in A for intersections with A.

Final Answer:
{c}