Symmetric Difference

Question

For finite sets the symmetric difference A Δ B is defined as the set of elements in exactly one of A or B; if n(A)=12, n(B)=9 and n(A ∩ B)=4, then n(A Δ B) equals which value?

RankGuru · Accepted Answer

The symmetric difference A Δ B collects elements in either set but not both, and its size is n(A) + n(B) - 2 n(A ∩ B), a direct consequence of inclusion-exclusion frequently used in JEE Advanced. Here n(A) + n(B) = 12 + 9 = 21, and twice the intersection is 2·4 = 8. Subtracting removes the shared elements once for each set, giving 21 - 8 = 13. Equivalently, the union has 21 - 4 = 17 elements, and removing the 4 common elements leaves 17 - 4 = 13 in exactly one set, confirming the formula by two routes. Option 17 reports the union size, which still includes common elements. Option 21 omits the overlap correction entirely. Option 8 mistakenly equals twice the intersection. Hence n(A Δ B) = 13. Plausibility check: the symmetric difference plus twice the intersection should reconstruct n(A)+n(B), and 13 + 8 = 21 matches exactly, validating the count of exactly-one-set elements.

Symmetric Difference

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