Let A, B, and C be subsets of a universal set U and suppose n(U)=100, n(A)=30, n(B)=30, n(C)=32, n(A∩B)=9, n(A∩C)=10, n(B∩C)=14, and n(A∩B∩C)=5.
Compute n(Ac∩Bc∩Cc) c.
70
64
39
14
10
89
I dont understand this. Please help. Thank you.
To solve this problem, we need to use a formula called the Inclusion-Exclusion Principle. This principle helps us count the number of elements in the union of multiple sets by subtracting the overlaps between the sets.
Here's how we can apply the principle in this case:
1. Start with the total number of elements in the universal set U, which is given as n(U)=100.
2. Let's calculate the number of elements in the complement of set A, denoted as Ac. Since n(A)=30, we can find n(Ac) by subtracting n(A) from n(U): n(Ac) = n(U) - n(A) = 100 - 30 = 70.
3. Similarly, we can find n(Bc) (the complement of set B) by subtracting n(B) from n(U): n(Bc) = n(U) - n(B) = 100 - 30 = 70.
4. Finally, we can find n(Cc) (the complement of set C) by subtracting n(C) from n(U): n(Cc) = n(U) - n(C) = 100 - 32 = 68.
Now, we need to find the number of elements in the intersection of the complements of A, B, and C, denoted as Ac∩Bc∩Cc.
To compute this, we can use the formula:
n(Ac∩Bc∩Cc) = n(U) - n(A∪B∪C).
Let's find n(A∪B∪C) using the Inclusion-Exclusion Principle:
n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C)
= 30 + 30 + 32 - 9 - 10 - 14 + 5
= 74.
Now, we can compute n(Ac∩Bc∩Cc):
n(Ac∩Bc∩Cc) = n(U) - n(A∪B∪C)
= 100 - 74
= 26.
Therefore, the value of n(Ac∩Bc∩Cc) is 26.