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Chapter 2: Q. 2.2 (page 52)

Prove the following relations:

IfE鈯�F,thenFc鈯�Ec

Short Answer

Expert verified

E鈯�F鈬�Fc鈯�Ec

Start by assumingE鈯�F.

Step by step solution

01

Given Information.

E鈯�F鈬�Fc鈯�Ec

02

Explanation.

Say thatE鈯�F, the definition of that is:

E鈯�F鈬�def(x鈭�E鈫�x鈭�Fand existsx鈭�Fsuch thatx鈭�E)

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

Ifx鈭�Fc

By the definition of complementx鈭�F.

x鈭�E鈫�x鈭�Fandx鈭�Fsoxis not inE.

xis a member of Ecand by the definition of a subsetFc鈯�Ec.

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

So by the same definitionFc鈯�Ec.

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