Question

Prove the following relations:

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

Short answer

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

Start by assuming$E\subset F$.

Step-by-step solution

Given Information.

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

Explanation.

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

$E\subset F\stackrel{\text{def}}{\iff }(x\in E\to x\in F\text{and exists}x\in F\text{such that}x\notin E)$

If$F^c=\varnothing$, 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$\left(1\right)$is a member of $E^c$and$x\notin F^c$.

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