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Chapter 2: Q. 2.2 (page 52)

Prove the following relations:

IfE⊂F,thenFc⊂Ec

Short Answer

Expert verified

E⊂F⇒Fc⊂Ec

Start by assumingE⊂F.

Step by step solution

01

Given Information.

E⊂F⇒Fc⊂Ec

02

Explanation.

Say thatE⊂F, the definition of that is:

E⊂F⇔def(x∈E→x∈Fand existsx∈Fsuch thatx∉E)

IfFc=∅, then Fcis a subset of all sets by convention.

Ifx∈Fc

By the definition of complementx∉F.

x∈E→x∈Fandx∉Fsoxis not inE.

xis a member of Ecand by the definition of a subsetFc⊆Ec.

and the element x∈Fsuch that x∉Efrom(1)is a member of Ecandx∉Fc.

So by the same definitionFc⊂Ec.

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