Let A, B, and C be subsets of a universal set U and suppose n(U) = 200, n(A) = 23, n(B) = 25, n(C) = 29, n(A ∩ B) = 7, n(A ∩ C) = 9, n(B ∩ C) = 14, and n(A ∩ B ∩ C) = 4. Compute:
(a) n[A ∩ (B ∪ C)]
(b) n[A ∩ (B ∪ C)c]
I suggest using a Venn diagram, all fields of the circles can be easily filled in.
Start with the intersection: n(A ∩ B ∩ C) = 4 , then do the doubles, etc
for b) , what is the meaning of c in n[A ∩ (B ∪ C)c] ?
To compute the values of n[A ∩ (B ∪ C)] and n[A ∩ (B ∪ C)c], we need to understand the concepts of union (∪), intersection (∩), and complement (c) of sets.
(a) Calculating n[A ∩ (B ∪ C)]:
Here, A ∩ (B ∪ C) refers to the elements that are common to both set A and the union of sets B and C. To find this, we can first compute B ∪ C, which represents the union of sets B and C. The union of two sets contains all the elements that are present in either set.
So, n(B ∪ C) = n(B) + n(C) - n(B ∩ C) = 25 + 29 - 14 = 40.
Next, we need to find the intersection of set A and the union of sets B and C. This indicates the elements that are present in both set A and the union of sets B and C.
n[A ∩ (B ∪ C)] = n(A ∩ (B ∪ C)) = n(A ∩ (B ∩ C)) = n(A ∩ B ∩ C) = 4.
Therefore, n[A ∩ (B ∪ C)] = 4.
(b) Calculating n[A ∩ (B ∪ C)c]:
Here, (B ∪ C)c refers to the complement of the union of sets B and C. The complement of a set contains all the elements of the universal set that are not present in the given set.
To find (B ∪ C)c, we need to subtract the elements in B ∪ C from the universal set U.
n(B ∪ C) = 40.
n[(B ∪ C)c] = n(U) - n(B ∪ C) = 200 - 40 = 160.
Now, we need to find the intersection between set A and the complement of the union of sets B and C.
n[A ∩ (B ∪ C)c] = n(A ∩ (B ∪ C)c) = n(A) - n(A ∩ (B ∪ C)) = n(A) - n[A ∩ (B ∪ C)] = 23 - 4 = 19.
Therefore, n[A ∩ (B ∪ C)c] = 19.