Q.In a survey it was found that 21 people liked product A, 26 liked product B and 29 liked product C. If 14 people liked products A and B, 12 people liked products C and A, 14 people liked products B and C and 8 liked all the three products. Find how many liked products C only.

This concept covers the CARDINALITY (counting) formulas that let you find the SIZE of a union or intersection without having to list out every element — essential for word problems involving surveys, exam results, or consumer preferences given only as totals and overlaps. For two sets: n(A∪B) = n(A) + n(B) − n(A∩B) — you add the two individual counts, then subtract the overlap once (since a direct sum of n(A)+n(B) would double-count every shared element). When A and B are disjoint, this simplifies to n(A∪B) = n(A)+n(B). Related identities let you isolate any one piece given the others, e.g. n(A∩B') = n(A) − n(A∩B) (the part of A outside B). For THREE sets, the inclusion-exclusion formula extends naturally: n(A∪B∪C) = n(A)+n(B)+n(C) − n(A∩B) − n(B∩C) − n(A∩C) + n(A∩B∩C) — each pairwise overlap, having been counted twice in the initial sum, is subtracted once; the triple overlap, having then been subtracted three times too many, is added back once. This formula (and its two-set version) is the standard tool for problems like 'how man …