## SectionB.1Introduction

Set theory is concerned with elements, certain collections of elements called sets and a concept of membership. For each element $x$ and each set $X\text{,}$ exactly one of the following two statements holds:

1. $x$ is a member of $X\text{.}$

2. $x$ is not a member of $X\text{.}$

It is important to note that membership cannot be ambiguous.

When $x$ is an element and $X$ is a set, we write $x\in X$ when $x$ is a member of $X\text{.}$ Also, the statement $x$ belongs to $X$ means exactly the same thing as $x$ is a member of $X\text{.}$ Similarly, when $x$ is not a member of $X\text{,}$ we write $x\notin X$ and say $x$ does not belong to $X\text{.}$

Certain sets will be defined explicitly by listing the elements. For example, let $X=\{a,b,d,g,m\}\text{.}$ Then $b\in X$ and $h\notin X\text{.}$ The order of elements in such a listing is irrelevant, so we could also write $X=\{g,d,b,m,a\}\text{.}$ In other situations, sets will be defined by giving a rule for membership. As examples, let $\posints$ denote the set of positive integers. Then let $X=\{n\in\posints:5\le n\le 9\}\text{.}$ Note that $6,8\in X$ while $4,10,238\notin X\text{.}$

Given an element $x$ and a set $X\text{,}$ it may at times be tedious and perhaps very difficult to determine which of the statements $x\in X$ and $x\notin X$ holds. But if we are discussing sets, it must be the case that exactly one is true.

###### ExampleB.1.

Let $X$ be the set consisting of the following $12$ positive integers:

\begin{align*} \amp 13232112332\\ \amp 13332112332\\ \amp 13231112132\\ \amp 13331112132\\ \amp 13232112112\\ \amp 13231112212\\ \amp 13331112212\\ \amp 13232112331\\ \amp 13231112131\\ \amp 13331112131\\ \amp 13331112132\\ \amp 13332112111\\ \amp 13231112131 \end{align*}

Note that one number is listed twice. Which one is it? Also, does $13232112132$ belong to $X\text{?}$ Note that the apparent difficulty of answering these questions stems from (1) the size of the set $X$ and (2) the size of the integers that belong to $X\text{.}$ Can you think of circumstances in which it is difficult to answer whether $x$ is a member of $X$ even when it is known that $X$ contains exactly one element?

###### ExampleB.2.

Let $P$ denote the set of primes. Then $35\notin P$ since $35= 5\times 7\text{.}$ Also, $19\in P\text{.}$ Now consider the number

\begin{equation*} n = 77788467064627123923601532364763319082817131766346039653933 \end{equation*}

Does $n$ belong to $P\text{?}$ Alice says yes while Bob says no. How could Alice justify her affirmative answer? How could Bob justify his negative stance? In this specific case, I know that Alice is right. Can you explain why?