Q. 2.1 - A First Course in Probability

Question

Prove the following relations:

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

Step-by-Step Solution

Answer

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

Step 1 Given Information.

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

Step 2 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 definition$\left(1\right)$,$E\subset E\cup F$