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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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