Operations On Sets

The inclusion-exclusion principle states n(A ∪ B) = n(A) + n(B) - n(A ∩ B), a foundational counting identity for finite sets in JEE Advanced. First compute the intersection: 33 = 18 + 23 - n(A ∩ B), so n(A ∩ B) = 41 - 33 = 8. Elements lying in exactly one set are those in the union but not in the intersection, counted by the symmetric-difference formula n(A)+n(B)-2 n(A ∩ B) = 18 + 23 - 2·8 = 41 - 16 = 25. Option 8 is just the intersection count, not the exactly-one count. Option 28 wrongly adds instead of subtracting twice. Option 31 ignores the overlap entirely. The governing pattern is the symmetric difference cardinality, a standard JEE set-algebra result. Plausibility check: exactly-one elements (25) plus twice the common elements (16) reconstruct 18+23=41, confirming internal consistency of the partition of the union. More broadly, the count of elements lying in exactly one of several sets emerges from a layered inclusion-exclusion in which odd-order overlaps are added and even-order overlaps subtracted. The decisive exam skill is translating verbal phrases like exactly one into the precise cardinality formula n(A)+n(B)-2n(A∩B) before any numbers are substituted, which prevents the common error of confusing this with the union count.