Answer to Question #167712 in Discrete Mathematics for Festus

Question

A sample poll of 135 voters revealed the following information in favour of three A ,B and C of a certain political party in Abia state. 60 were in favour of B; 23 were in favour of A alone; 17 were in favour of C alone; 10 were sitting on the fence; 55 were in favour of exactly two candidates; 61 were in favour of exactly one candidate; 46 were in favour of candidate A and at least one other candidate.
How many people were in favour of
(a) both B and C
(b) A and B only?
(c) A or C

Answer 1

$$|S|=135, |A\cup B \cup C|=135-10=125$$

$$|A\ alone|=23, |C\ alone|=17,$$

$$|B\ alone|=61-23-17=21$$

$$y+z+w+21=60$$

$$x+y+z=55$$

$$x+y+w=46$$

$$61+55+w=125$$

Then

$$w=9$$

$$x+y=37$$

$$y+z=30$$

$$z=18$$

$$x=25,y=12,z=18,w=9$$

(a)

$$|B\cap C|=18+9=27$$

(b) A and B only?

$$|A\ and\ B\ only|=12$$

(c)

$$|A\cup C|=23+17+25+9=74$$