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If n(A) = 20, n(B) = 30 and n(A union B) = 45, then n(A intersection B) is:
5
10
15
20
5
Simply subtract the sum of individual sets from the union: (20+30)тИТ45=5.
Number of elements in set A is 20, set B is 30, and the union of sets A and B is 45.
n(AтИкB)=n(A)+n(B)тИТn(AтИйB)
Simply subtract the sum of individual sets from the union: (20+30)тИТ45=5.
Students often add the sets and subtract the intersection from the total, instead of using the inclusion-exclusion principle to isolate the intersection.
State the formula
Use the Principle of Inclusion-Exclusion for two sets.
n(AтИкB)=n(A)+n(B)тИТn(AтИйB)
Rearrange for intersection
Isolate the intersection term n(AтИйB) on one side of the equation.
n(AтИйB)=n(A)+n(B)тИТn(AтИкB)
Substitute given values
Substitute n(A)=20, n(B)=30, and n(AтИкB)=45 into the equation.
n(AтИйB)=20+30тИТ45
Calculate the result
Perform the arithmetic: 50тИТ45=5.
n(AтИйB)=5
A is correct because the application of the inclusion-exclusion principle yields a result of 5.
This principle is foundational for Probability, specifically for calculating the probability of the union of two non-mutually exclusive events: P(AтИкB)=P(A)+P(B)тИТP(AтИйB).