Find the number of elements in a1 ∪ a2 ∪ a3 if there are 100 elements in a1, 1000 in a2, and 10,000 in a3 if
a) a1 ⊆ a2 and a2 ⊆ a3.
b) the sets are pairwise disjoint.
c) there are two elements common to each pair of sets and one element in all three sets.

(a) 1000

(b) 11100

(c) 11095.

Step-by-step explanation:  

(a) If A1 is a subset of A2 and A2 is a subset of A3, then all the elements of A1 are in A2 and all the elements of A2 are in A3.

Then, n(A1 n A2) = 100, n(A2 n A3) = 1000 , n(A1 n A3) = 100 and n(A1 n A2 n A3) = 100.

So,  we get

(b) If the sets are pairwise disjoint, then

n(A1 n A2) = n(A2 n A3) = n(A1 n A3) = n(A1 n A2 n A3) = 0.

So, we get

(c) If  there are two elements common to each pair of sets and one element in all three sets, then

n(A1 n A2) = 2,  n(A2 n A3) = 2, n(A1 n A3) = 2 and n(A1 n A2 n A3) = 1.

So, we get

- (2, -7)

- lines r and s and lines t and u must be parallel