Symmetric Difference
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?
Select the correct option:
Solution
13
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.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- symmetric difference
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
13
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.
This easy difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of symmetric difference. It appeared in the 2025 exam.
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