Answer to Question #277201 in Discrete Mathematics for Ram

Question

2. A survey on subjects being taken by 250 college students in Candon City revealed the following information.

Subject Number of students taking the subject

Mathematics 145

Filipino 90

English 88

Math and Filipino 25

Filipino and English 38

Math and English 59

Math, Filipino and English 15

How many did not take any of the three subjects?

How many took Math as their only subject? English only? Filipino only?

How many were taking Math and Filipino but not English?

How many were taking Filipino and English but not Math?

Answer 1

$$N(M\cup F\cup E)=N(M)+N(F)+N(E)$$

$$-N(M\cap F)-N(M\cap E)-N(F\cap E)$$

$$+N(M\cap F\cap E)=145+90+88$$

$$-25-38-59+15=216$$

a)

$$N(not\ any)=N(U)-N(M\cup F\cup E)$$

$$=250-216=34$$

b)

$$N(only\ one)=N(only\ M)$$

$$+N(only\ F)+N(only\ E)$$

$$N(only\ M)=N(M)-N(M\cap F)-N(M\cap E)$$

$$+N(M\cap F\cap E)=145-25-59+15=76$$

$$N(only\ F)=N(F)-N(M\cap F)-N(F\cap E)$$

$$+N(M\cap F\cap E)=90-25-38+15=42$$

$$N(only\ E)=N(E)-N(M\cap E)-N(F\cap E)$$

$$+N(M\cap F\cap E)=88-38-59+15=6$$

$$N(only\ one)=76+42+6=124$$

c)

$$N((M\cap F)-E)=N(M\cap F)-N(M\cap F\cap E)$$

$$=25-15=10$$

d)

$$N((F\cap E)-M)=N(F\cap E)-N(M\cap F\cap E)$$

$$=38-15=23$$