Prove the following relations:IfE⊂F, then Fc ⊂Ec

E⊂F⇒Fc⊂EcStart by assumingE⊂F.

Q. 2.2 - A First Course in Probability

Prove the following relations:

If $E \subset F,$ then $F^{c} \subset E^{c}$

Verified

Answer

$E \subset F \Rightarrow F^{c} \subset E^{c}$

Start by assuming $E \subset F$ .

1Step 1 Given Information.

$E \subset F \Rightarrow F^{c} \subset E^{c}$

2Step 2 Explanation.

Say that $E \subset F$ , the definition of that is:

$E \subset F \overset{def}{\Leftrightarrow} (x \in E \to x \in F and exists x \in F such that x \notin E)$

If $F^{c} = \emptyset$ , then $F^{c}$ is a subset of all sets by convention.

If $x \in F^{c}$

By the definition of complement $x \notin F$ .

$x \in E \to x \in F$ and $x \notin F$ so $x$ is not in $E$ .

$x$ is a member of $E^{c}$ and by the definition of a subset $F^{c} \subseteq E^{c}$ .

and the element $x \in F$ such that $x \notin E$ from $(1)$ is a member of $E^{c}$ and $x \notin F^{c}$ .

So by the same definition $F^{c} \subset E^{c}$ .