Seemingly simple meaning of union and intersection of sets (events)

Question

Let $E, F$ and $G$ be three events. Explain the meaning of the relation $E\cup F\cup G = G.$

The answer is: “If $E$ or $F$ occurs, then $G$ occurs.” Why?

Answer 1

Think of an event as a set of outcomes of an experiment.

Then an event $E$ occurs if the outcome of the experiment is an element of $E$.

If $E\cup F\cup G=G$ then all outcomes in $E\cup F$ are also outcomes in $G$.

So if the outcome is in $E$ or in $F$ then it is in $G$ as well.

Translated: if $E$ occurs or $F$ occurs then $G$ occurs.

Answer 2

Let $S$ be the sample space.

Suppose $E\cup F\cup G=G$.

If $E$ occurs at sample point $x$, then \begin{align*} &x\in E\\[4pt] \implies\;&x\in E\cup F \cup G\\[4pt] \implies\;&x\in G\\[4pt] \implies\;&G\;\text{occurs}\\[4pt] \end{align*} Thus, whenever $E$ occurs, $G$ occurs.

By analogous reasoning, whenever $F$ occurs, $G$ occurs.

Hence whenever $E$ or $F$ occurs, $G$ occurs.

Conversely, suppose whenever $E$ or $F$ occurs, $G$ occurs.

We want to show $E\cup F\cup G=G$

Since $G$ is a subset of the $\text{LHS}$, it follows that $G\subseteq E\cup F\cup G$.

It remains to show the reverse inclusion $E\cup F\cup G\subseteq G$.

Let $x\in E$.

If $x$ occurs, then $E$ occurs, hence, by assumption, $G$ occurs, so $x\in G$.

Thus, $x\in E$ implies $x\in G$, so $E\subseteq G$.

By analogous reasoning $F\subseteq G$.

And of course, $G\subseteq G$.

Hence, $E\cup F\cup G\subseteq G$.

Since we have both inclusions, we get $E\cup F\cup G=G$.

Answer 3

Given $E \cup F \cup G = G$

This implies G is a bigger set than $E \cup F $ . Also further $E,F$ are subsets of $G$ .

Thinking in terms events $G$ is a bigger event than $E$ and $F$ . Since $E$ and $F$ are subsets of G . The event of $F$ is an event in $G$ also (For example take $F$ as getting 2 or 4 when a die is rolled and $G$ as getting a even number when a die is rolled) .Hence If $F$ occurs, $G$ occurs.

Answer 4

Note that $A \cup B = B$ is equivalent to $A \subset B$ with the interpretation that if A occurs then B occurs. Also $A \cup B$ has the interpretation A or B occurs. Thus you have $E \cup F \subset G$ which reads as if E or F occurs then G occurs.