Question

For the following questions assume that A,B,C ⊆ U.

1. Suppose A = {1,2,3,a,b} and B = {2,3,5,b}.

   (a) Find A ∪ B.

2. (b) Find A ∩ B.

1. Suppose n(A) = 30, n(B) = 60, n(C) = 20, A and C are disjoint. What is the smallest that n(U) can be? Explain.

2. Ifn(U)=100,n(A)=40,andn(B)=50. Findn(A′∪B′)ifn(A′∩B′)=30.

3. Suppose a pizza parlor has a deal for two-topping pizzas. There are 4 types of crust, 3 types of cheese, and 12 types of toppings that are eligible for this deal. If the toppings for the pizza must be distinct, how many two-topping pizzas are possible?

Answer 1

A)

A U B

Means all the elements of A and B (avoid repetitions)

A U B = {1,2,3,5,a,b}

B)

A intersection B

{2,3,b}

That is common elements of A and B

2)

Suppose n(A) = 30, n(B) = 60, n(C) = 20, A and C are disjoint. What is the smallest that n(U) can be? Explain.

Answer)

A and C are disjoint

So n(U) = maximum number of A + C and B

As repetition means nothing in sets

So, smallest that n(U) can be is 60

3)

Ifn(U)=100,n(A)=40,andn(B)=50. Findn(A′∪B′)ifn(A′∩B′)=30.

Answer)

N(A' intersection B') = 30

(A' intersection B') = (A U B)'

So, n(A U B)' = 30

We know that

n(A U B)' = n(U) - n(AUB) =

30 = 100 - n(AUB)

n(A U B) = 70

We also know that

n(A U B) = n(A) + n(B) - n(A intersection B)

70 = 40 + 50 - n(A intersection B)

n(A intersection B) = 20

We need to find n (A' U B') which is = n(A intersection B)' = U - n(A intersection B)

So, n(A' U B') = 100 - 20 = 80