Answer to Question #290982 in Discrete Mathematics for Ayya

Question

A group of students study at least one of the following subjects:biology, chemistry and physics. 36 of them study chemistry, 39 study biology and 41 study physics.15 study physics. 9 study all the three subjects. @draw a Venn diagram to illustrate this information.(b)from your diagram find:(1) the total number of students in the group.(2)the number of students who study only one subject.@the universal set U={2,3,5,7},p={2,5}and Q={5,7}find:(1)(PnQ),(2)P'UQ'. State the relationship between (1)and(2)(b)in a class of 50 students, 30 offer economics,17 offer government and 7 offer neither economics nor government. How many students offer both subjects.

Answer 1

1.

(a)

$$N(C)=c+x+z+9$$

$$N(B)=b+x+y+9$$

$$N(P)=p+y+z+9$$

36 study chemistry:

$$c+x+z+9=36$$

39 study biology:

$$b+x+y+9=39$$

41 study physics

$$p+y+z+9=41$$

15 study only physics

$$p=15$$

There is not enough information to solve the problem.

2.

(a)

$$U=\{2,3,5,7\},P=\{2,5\},Q=\{5,7\}$$

(1)

$$(P\cap Q)=\{5\}$$

(2)

$$P'=\{3,7\}, Q'=\{2,3\}$$

$$(P'\cup Q')=\{2,3,7\}$$

$$(P\cap Q)=(P'\cup Q')'$$

(b)

$$N(E\cup G)=N(U)-N(E'\cap G')$$

$$=50-7=43$$

$$N(E\cup G)=N(E)+N(G)-N(E\cap G)$$

$$N(E\cap G)=30+17-43=4$$

4 students offer both subjects.