Question

Prove the following relations:

$EF\subset E\subset E\cup F$

Short answer

Use the definition of a subset and logical operations$EF\subset E\subset E\cup F$.

Step-by-step solution

Given Information.

$EF\subset E\subset E\cup F$

Explanation.

$EF\subseteq E\subseteq E\cup F$

If$EF=\varnothing$, then $EF=\varnothing$is a subset of all sets by convention.

If$x\in EF$

$$\begin{gathered} \text{By the definition of intersection}x\in E\text{and}x\in F \\ \to x\in E \end{gathered}$$

$x$is a member of $E$and by the definition of a subset

$A\subseteq B\stackrel{\text{def}}{\iff }x\in A\to x\in B$

$EF\subseteq E$The same procedure for $E\subset E\cup F$If$E=\varnothing$then $E$is a subset of all sets by convention.

If$x\in E$

$\to x\in E\text{or}x\in F$

By the definition of a union$x\in E\cup F$

By the definitionlocalid="1649042050551" $\left(1\right)$,$E\subset E\cup F$