Answer to Question #279119 in Discrete Mathematics for Freedi

Question

A pet store keeps track of the purchases of customers over a fours hour period. The store manager classifies purchases as containing a dog product, a cat product, a fish product, or product for a different kind of pet. He found!

83 purchased a dog product

101 purchased a cat product

22 purchased a fish product

31 purchased a dog and a cat product

8 purchased a dog and a fish product

10 purchased a cat and a fish product

6 purchased a dog, a cat, and a fish product

34 purchased a product for a pet other than a dog, cat, or fish.

Draw a Venn diagram to find that:

(1) How many purchases were for a dog product only?

(ii) How many purchases were for a cat product only?

How many purchases were for a dog or a fish product?

(iv) How many purchases were there in total?

Answer 1

83 purchased a dog product$\Rightarrow n(dog)=83-25-6-2=50$

101 purchased a cat product$\Rightarrow n(cat)=101-25-6-4=66$

22 purchased a fish product$\Rightarrow n (fish)=22-6-4-2=10$

31 purchased a dog and a cat product$\Rightarrow n(cat \cap dog)=31-6=25$

8 purchased a dog and a fish product$\Rightarrow n( fish\ \cap dog)=8-6=2$

10 purchased a cat and a fish product$\Rightarrow n(cat\ \cap fish)=10-6=4$

6 purchased a dog, a cat, and a fish product $\Rightarrow n(cat\ \cap fish\ \cap dog)=6$

34 purchased a product for a pet other than a dog, cat, or fish $\Rightarrow n(cat\ \cap fish\ \cap dog)'=34$

Venn diagram on the given information is as follows:

Now from the venn diagram, we can answer these questions.

(i) How many purchases were for a dog product only?

Ans: 50

(ii) How many purchases were for a cat product only?

Ans: 66

(iii) How many purchases were for a dog or a fish product?

Ans: 50+2+10 = 62

(iv) How many purchases were there in total?

Ans: 50+2+10+25+6+4+66+34 = 197