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Examoogle • User • info@examoogle.com • EE-2024-8821
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If X and Y are two sets such that X ∪ Y has 56 elements, X has 45 elements and Y has 33 elements, how many elements does X ∩ Y have ?
13
22
19
25
22
Simply add the two individual sets and subtract the union: (45 + 33) - 56. The result is 78 - 56 = 22.
Total elements in union n(X ∪ Y) = 56, elements in set X n(X) = 45, elements in set Y n(Y) = 33.
n(X∪Y)=n(X)+n(Y)−n(X∩Y)
Simply add the two individual sets and subtract the union: (45 + 33) - 56. The result is 78 - 56 = 22.
Students often add the individual sets (45 + 33 = 78) and mistake this for the answer, forgetting to subtract the union to account for the overlapping elements.
Identify given values
We are given the number of elements in set X as n(X)=45, set Y as n(Y)=33, and their union n(X∪Y)=56.
n(X)=45,n(Y)=33,n(X∪Y)=56
Apply set theory formula
Use the inclusion-exclusion principle to find the intersection n(X∩Y).
n(X∩Y)=n(X)+n(Y)−n(X∪Y)
Calculate the result
Substitute the given values into the formula: 45+33−56.
n(X∩Y)=78−56=22
B is correct because the application of the inclusion-exclusion principle results in 22 elements being common to both sets.
This concept is fundamental for Data Interpretation questions involving Venn diagrams and is frequently tested in probability problems where calculating overlapping sample spaces is required.