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Chapter 2: Q. 2.7 (page 52)

Use Venn diagrams

(a)to simplify the expressions (E∪F)(E∪Fc);

(b)to prove DeMorgan’s laws for eventsEandF. [That is, prove(E∪F)c=EcFc, and (EF)c=Ec∪Fc]

Short Answer

Expert verified

By using the commutative, distributive, and associative properties of(∪,∩)

a)E

b)EF

c)F∪EG

Step by step solution

01

Given Information.

(a)expressions(E∪F)(E∪Fc);

(b)(E∪F)c=EcFc,and(EF)c=Ec∪Fc.

02

Part (a) Explanation.

a) Simplification of (E∪F)E∪Fc

(E∪F)E∪Fc=(E∪F)E∪(E∪F)Fc

=E∪(E∪F)Fc

=E∪EFc∪FFc

=E∪EFc∪FFc

=E∪EFc∪∅

=E∪EFc

=E

03

part (b) Explanation.

b) Simplification of (E∪F)Ec∪FE∪Fc

(E∪F)Ec∪FE∪Fc=(E∪F)E∪FcEc∪F

=(E∪F)E∪FcEc∪F

=EEc∪F

=EEc∪EF

=∅∪EF

=EF

04

part (c) Explanation.

c) Simplification of (E∪F)(F∪G)

(E∪F)(F∪G)=(E∪F)F∪(E∪F)G

=F∪(E∪F)G

=F∪EG∪FG

=F∪FG∪EG

=F∪EG

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Most popular questions from this chapter

An urn contains 5red, 6blue, and 8green balls. If a set of 3balls is randomly selected, what is the probability that each of the balls will be

(a) of the same color?

(b) of different colors? Repeat under the assumption that whenever a ball is selected, its color is noted and it is then replaced in the urn before the next selection. This is known as sampling with replacement .

Five people, designated as A,B,C,D,E, are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that

(a) there is exactly one person between Aand B?

(b) there are exactly two people between Aand B?

(c) there are three people between AandB?

An urn contains nwhite and mblack balls, wherenandmare positive numbers.

(a) If two balls are randomly withdrawn, what is the probability that they are the same color?

(b)If a ball is randomly withdrawn and then replaced before the second one is drawn, what is the probability that the withdrawn balls are the same color?

(c)Show that the probability in part (b)is always larger than the one in part (a).

Prove thatP(EFc)=P(E)−P(EF).

Given 20 people, what is the probability that among the 12 months in the year, there are 4 months containing exactly 2 birthdays and 4 containing exactly 3 birthdays?
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