Sets

In mathematics we build new objects and theorems from previous ones, with ever increasing complexity. However, we have to start somewhere in math, and for much of modern mathematics we start with set theory.

Sets are often thought of as primitive notions. This means they can’t be defined from other concepts, and act as a bedrock for building more complex mathematical objects. We build set theory from two primitive notions, sets and containment.

Definition

A set is a collection of distinct elements.

This definition is most likely unsatisfying. We have said nothing on the nature of what it means to be a collection or what it means to be an element. However, it is good enough for the purposes of this chapter, and once one has an intuitive understanding of a set one may look to axiomatic set theory for more concrete definitions and answers. For now though, lets examine what is meant by an element.

Definition

An element is an object contained by a set.

Holy circular reasoning! Instead of getting too philosophical on the nature of these sets and elements, lets look at some examples.

Notation

We usually denote a set by a capital letter A,B,C,\dots and an element of a set by a lowercase letter a,b,c etc.

We very commonly use brace notation for writing sets, where we denote a set A by it’s elements listed out, for example

A = \{a,b,c\}

or when a set contains way more elements than could be listed, for example the set of all integers, we write

B = \{1,2,3,\dots\}.

Notation

If an element a is contained by a set A, we write

a \in A.

If an element a is not contained by a set A, we write

a \not\in A.

A set is completely determined by it’s elements. This means if two sets A,B have exactly the same elements, they are the exact same set, and we may say A = B. This is not a trivial declaration, and is an axiom known as the axiom of extensionality.

Principle of Extensionality

Two sets are equal if and only if they have the same elements.

Another point to note is that a set only contains distinct elements. An element is either belongin to a set or not, and may not be contained twice so to speak. Additionally, the way we order elements in a set is irrelevant, as two sets are equal if they contain the same elements, so

\{a,b\} = \{b,a\}

Example

A = \{a,b,c\} \quad B = \{b,a,a,a\}

A is an example of a set as each element only occurs once, but B is not a set as a occurs multiple times. Sometimes it may be convenient to just ignore an element if it occurs multiple times, such that B = \{b,a,a,a\} = \{b,a\}

but realistically we shouldn’t encounter this situation.

Types of Sets

There are special types of sets which come in very handy for our further study.

Definition

The empty set \varnothing is the set containing no elements:

\varnothing := \{\}.

The empty set is unique, as any two sets with no elements will be equivalent. We also have the opposite of an empty set, the infinite set.

Definition

The set \mathbb{N} is the set of all natural numbers:

\mathbb{N} := \{0,1,2,3,\dots\}.