Union cardinality via inclusion–exclusion (given |S|, |T|, |S ∩ T|): If |S| = 21, |T| = 32, and |S ∩ T| = 11, find |S ∪ T|.

Introduction / Context:The cardinality of a union of two finite sets is found by inclusion–exclusion: add individual sizes and subtract the intersection to avoid double-counting shared elements. This is a fundamental counting identity.

Given Data / Assumptions:

• |S| = 21
• |T| = 32
• |S ∩ T| = 11

Concept / Approach:Apply |S ∪ T| = |S| + |T| − |S ∩ T|. This ensures elements common to S and T are counted once overall.

Step-by-Step Solution:|S ∪ T| = 21 + 32 − 11 = 42

Verification / Alternative check:Check extremes: if disjoint, union would be 53; since 11 overlap, reduce by 11 to get 42—consistent.

Why Other Options Are Wrong:52 ignores subtracting the overlap; 32 is one set only; “None of these” is unnecessary because 42 is exact.

Common Pitfalls:Adding sizes without subtracting the intersection, leading to overcounts.

Final Answer:42