2.1 Introduction to Sets

In modern mathematics, set theory is important as a foundation from which all of mathematics can be derived. All of the mathematical objects we deal with in analysis will be defined in terms of sets.

Definition 2.1.

A set is an unordered collection of objects. If an object $x$ is contained in a set $A$ , we call $x$ an element or member of $A$ and write $x\in A$ . Likewise if $x$ is not contained in $A$ we can say $x$ does not belong to $A$ and write $x\notin A$ . Sets are notated with curly braces {} around a comma-separated list of the elements.

The elements of a set can be any mathematical object but most sets we will encounter will be sets of numbers, such as $\mathbb{N}=\{1,2,3,...\}$ , the set of natural numbers¹¹ 1 Note that in this text, $0\notin\mathbb{N}$ . Generally there is no strong convention on whether 0 is included or not so it is important to check with every text.; or $\mathbb{Z}=\{...,{-2}{-1},0,1,2,...\}$ , the set of integers (‘…’ means ‘and so on forever’).

Definition 2.2.

There is a set which has no elements, and this set is unique. We call it the empty set and denote it $\emptyset$ . By definition for every object $x$ we have $x\notin\emptyset$ . If we say a set $A$ is non-empty, then $\exists$ some object $x$ such that $x\in A$ .

It is useful to have the ability to define a set in terms of some kind of predicate, for example let $E$ be ‘the set of all even integers’. We can write this in a more compact and unambiguous way as $E=\{2k:k\in\mathbb{Z}\}$ (The colon, sometimes replaced with $|$ , is read as ‘such that’). This is known as set-builder notation.

Definition 2.3.

Two sets $A$ and $B$ are equal $\iff$

\forall x\in A,x\in B\ \text{and}\ \forall y\in B,y\in A.

Then we write $A=B$ . In other words, both sets must have the same elements, but note that multiplicity and order do not matter, for example $\{1,2,3\}=\{3,2,1,1\}$ . Two sets defined using set-builder notation are equal if and only if their predicates are equivalent.

Definition 2.4.

Let $A$ be a set. Then a set $B$ is a subset of $A\iff$

\forall x\in B,x\in A.

Then we write $B\subseteq A$ . Note that every set has two trivial subsets, itself and $\emptyset$ .

Definition 2.5.

Let $A$ be a set. Then a set $B$ is a proper subset of $A\iff$

\forall x\in B,x\in A\ \text{and}\ A\neq B.

Then we write $B\subset A$ .