Let A and B be subsets of a universal set U and suppose n(U) = 390, n(A) = 130, n(B) = 90, and n(A B) = 40. Find the number of elements in the set.
n(Ac n Bc)
390-(130+90-40) = ____
Ac ∩ Bc = (A U B)c
in other words, everything in U which is not in A or B or both.
To find the number of elements in the set n(Ac n Bc), we first need to understand what this set represents.
The set Ac represents the complement of set A, which consists of all the elements in the universal set U that are not in set A. Similarly, the set Bc represents the complement of set B, which consists of all the elements in the universal set U that are not in set B.
So, n(Ac n Bc) represents the number of elements that are in both sets Ac and Bc.
To find n(Ac), we can 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) = 390 - 130 = 260.
Similarly, to find n(Bc), we can subtract the number of elements in set B from the total number of elements in the universal set U:
n(Bc) = n(U) - n(B) = 390 - 90 = 300.
Now, we need to find the number of elements that are in both sets Ac and Bc. Since Ac and Bc are complements of sets A and B, respectively, any element that is not in A will be in Ac, and any element that is not in B will be in Bc. Therefore, to find n(Ac n Bc), we can subtract the number of elements in the intersection of A and B from the total number of elements in the universal set U:
n(Ac n Bc) = n(U) - n(A B) = 390 - 40 = 350.
So, the number of elements in the set n(Ac n Bc) is 350.