From "Discrete Mathematics with Graph Theory"

by Goodaire and Parmenter

Page 191 # 5&6b

5. One day a candy store owner is swamped by 15 children. He notices that 10 purchased popsicles, seven purchased gum, twelve purchased candy bars, five purchased popsicles and gum, six purchased popsicles and candy bars, and two purchased gum and candy bars. Why is the owner upset?

Attempt:

P = popsicles

G= gum

C = candy bars

I'm not really sure how to go about this..

10P ∪ 7G ∪ 12CB ∪ 5(P∪G) ∪ 6(P∪CB) ∪ 2(G∪CB)

then idk what else to do.

6b. In a group of 97 students, the number taking English is twice the number taking math. 53 students take exactly 1 of these subjects and 15 are taking either course. How many students are taking math? How many are taking English?

For this is am totally lost.