Answer to Question #203028 in Discrete Mathematics for lucy

Question

There were 100 students in the library who responded to how they completed their research paper. • 18 students only used the periodicals. • 29 students used the web and books. • 15 students used books, the web, and periodicals. • 40 students used books and periodicals. • 20 used the web and periodicals. • 60 students used books. • 7 students did not use the web, nor books, nor the periodicals. a) Represent this information with a Venn diagram. (8 marks) b) How many students only used the web in their research? (2 marks) c) How many students used books or periodicals?

Answer 1

a)

$$N(P\cap B'\cap W')=18$$

$$N(B\cap W)=29$$

$$N(B\cap W\cap P)=15$$

$$N(B\cap W\cap P')=N(B\cap W)-N(B\cap W\cap P)$$

$$=29-15=14$$

$$N(B\cap P)=40$$

$$N(B\cap P\cap W')=N(B\cap P)-N(B\cap P\cap W)$$

$$=40-15=25$$

$$N(W\cap P)=20$$

$$N(W\cap P\cap B')=N(W\cap P)-N(B\cap P\cap W)$$

$$20-15=5$$

$$N(B)=60$$

$$N(B\cap P'\cap W')=N(B)-N(B\cap W)-N(B\cap P)$$

$$+N(B\cap W\cap P)=60-29-40+15=6$$

$$N(B'\cap W' \cap P')=7$$

$$N(W\cap B'\cap P')=100-N(B'\cap W' \cap P')$$

$$-N(P\cap B'\cap W')-N(W\cap P\cap B')$$

$$=100-7-60-18-5=10$$

b)

$$N(W\cap B'\cap P')=10$$

c)

$$N(B\cup P)=N(B\cap W'\cap P')+N(P\cap B'\cap W')$$

$$+N(B\cap P)=6+18+40=64$$