Let A, B, And C Be Sets In A Universal Set U.

Let A, B, and C be sets in a universal set U. We are given n(U) = 73, n(A) = 40, n(B) = 31, n(C) = 32, n(A ∩ B) = 19, n(A ∩ C) = 19, n(B ∩ C) = 17, n(A ∩ B ∩ C) = 9. Find the following values.

Let us analyze the problem systematically. Given the sizes of the sets and their intersections within a universal set U of size n(U) = 73, the goal is to determine the cardinality of the complement of the union (part a) and the intersection of the complements (part b).

Problem Breakdown and Solution

(a) Find n((A ∪ B ∪ C)^c)

The complement of the union of A, B, and C, denoted as (A ∪ B ∪ C)^c, consists of all elements in the universal set U that are not in A, B, or C.

Recall the basic set theory relation:

• n((A ∪ B ∪ C)^c) = n(U) - n(A ∪ B ∪ C)

Therefore, to find n((A ∪ B ∪ C)^c), we first need to compute n(A ∪ B ∪ C).

Calculating n(A ∪ B ∪ C): Inclusion-Exclusion Principle

The principle states:

n(A ∪ B ∪ C) = n(A) + n(B) + n(C) - n(A ∩ B) - n(A ∩ C) - n(B ∩ C) + n(A ∩ B ∩ C)

Plugging in the known values:

• n(A) = 40
• n(B) = 31
• n(C) = 32
• n(A ∩ B) = 19
• n(A ∩ C) = 19
• n(B ∩ C) = 17
• n(A ∩ B ∩ C) = 9

Now, compute:

n(A ∪ B ∪ C) = (40 + 31 + 32) - (19 + 19 + 17) + 9 = 103 - 55 + 9 = 57

Next, determine n((A ∪ B ∪ C)^c):

n((A ∪ B ∪ C)^c) = n(U) - n(A ∪ B ∪ C) = 73 - 57 = 16

Answer for (a):

n((A ∪ B ∪ C)^c) = 16

(b) Find n(A^c ∩ B^c ∩ C)

Note that:

A^c ∩ B^c ∩ C = (A ∪ B ∪ C)^c ∩ C

But, since (A ∪ B ∪ C)^c contains all elements not in A, B, or C, and C is itself the set of elements in C, the intersection simplifies to the elements outside A or B but inside C? Not quite—this needs clearer interpretation.

Actually, the set A^c ∩ B^c ∩ C includes elements that are outside A and B simultaneously, but are in C. Equivalently, this is the subset of C that is in neither A nor B:

A^c ∩ B^c ∩ C = C \ (A ∪ B)

Hence, the number of elements in A^c ∩ B^c ∩ C is:

n(C) - n(C ∩ (A ∪ B))

We know n(C) = 32, so we need n(C ∩ (A ∪ B)).

Calculating n(C ∩ (A ∪ B)):

n(C ∩ (A ∪ B)) = n((C ∩ A) ∪ (C ∩ B))

Applying the inclusion-exclusion principle:

n((C ∩ A) ∪ (C ∩ B)) = n(C ∩ A) + n(C ∩ B) - n(C ∩ A ∩ B)

Given data:

• n(C ∩ A) = 19
• n(C ∩ B) = 17
• n(A ∩ B ∩ C) = 9

Calculating:

n(C ∩ (A ∪ B)) = 19 + 17 - 9 = 27

Now, the number of elements in A^c ∩ B^c ∩ C is:

n(C) - n(C ∩ (A ∪ B)) = 32 - 27 = 5

Answer for (b):

n(A^c ∩ B^c ∩ C) = 5

Conclusion

In summary, the calculations reveal that 16 elements in the universal set are outside of the union of A, B, and C, and 5 elements are in C but outside of A and B simultaneously. These insights are critical in understanding the structure of overlapping sets within the universal context and demonstrate the applicability of inclusion-exclusion principles in solving complex set problems.

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