Let A and B be subsets of a universal set U and suppose n(U)=210, n(A)=100, n(B)=60, and n(A∩B)=30. Compute n(Ac∩Bc).
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150
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200
Please help. Thank you
To calculate the number of elements in the intersection of the complement of A and the complement of B, denoted as Ac∩Bc, we need to understand a few concepts.
First, let's define the complement of a set. The complement of a set A, denoted as Ac, refers to all the elements in the universal set U that are not in A.
Second, we need to understand the property of intersection. The intersection of two sets A and B, denoted as A∩B, refers to all the elements that are common to both sets.
Now, let's break down the problem.
Given information:
n(U) = 210 (the number of elements in the universal set U)
n(A) = 100 (the number of elements in set A)
n(B) = 60 (the number of elements in set B)
n(A∩B) = 30 (the number of elements in the intersection of sets A and B)
To calculate n(Ac∩Bc), we need to find the number of elements that are in the intersection of the complements of sets A and B.
To find n(Ac), first, we need to subtract the number of elements in set A from the total number of elements in the universal set U:
n(Ac) = n(U) - n(A) = 210 - 100 = 110.
Similarly, to find n(Bc), we subtract the number of elements in set B from the total number of elements in the universal set:
n(Bc) = n(U) - n(B) = 210 - 60 = 150.
Now that we have the values for n(Ac) and n(Bc), we can find n(Ac∩Bc) by calculating the intersection of Ac and Bc:
n(Ac∩Bc) = n(Ac) ∩ n(Bc) = 110 ∩ 150.
To calculate the intersection, we need to find the smaller value between n(Ac) and n(Bc), which is 110.
Hence, n(Ac∩Bc) = 110.
Therefore, the correct option is 110.