Associativity of union over three sets in a finite universe: Given U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}. Compute (A ∪ B) ∪ C.

Introduction / Context:Union is associative and idempotent. When two sets already cover the universe U, adding a third does not change the result. We verify elementwise here.

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:Compute A ∪ B first; then union with C. Since A ∪ B already equals U, any further union remains U.

Step-by-Step Solution:A ∪ B = {a, b, c} ∪ {c, d, e, f} = {a, b, c, d, e, f} = U(A ∪ B) ∪ C = U ∪ C = U

Verification / Alternative check:Directly, A ∪ B ∪ C contains all symbols in U; there is no symbol outside U, hence the union equals U.

Why Other Options Are Wrong:A, B, or C are proper subsets of U; A ∪ C still misses f; only U contains every element.

Common Pitfalls:Accidentally intersecting instead of uniting; forgetting union is associative so parentheses do not alter the final set here.

Final Answer:U