Sets

Definition: Set

A set is a collection of well-defined objects, called elements.

NOTATION

The notation $a\in A$ means that $a$ is an element of the set $A$.

The notation $a\notin A$ means that $a$ is not an element of the set $A$.

NOTE

• A set can contain pretty much anything - numbers, letters, cars, sentences, people, colours and even other sets.
• A set can contain either finitely many or infinitely many elements.

Definition: Singleton

A singleton is a set with exactly one element.

Specifying Sets

There are three main ways to represent and define sets.

The descriptive form uses words to describe a set. For example, the set $S$ is the set of all odd natural numbers which are less than 12.

The set-builder form defines a set by specifying a condition that all of its members satisfies and looks like this:

$$\text{set}=\{\text{ placeholder }|\text{ condition }\}$$

The placeholder is simply there so you can use it to more easily write the condition. The | character can be read as “such that”. For example, specifying the aforementioned set $S$ using set-builder notation will look like the following.

$$S=\{s|s\text{ is an odd number and }0<s<12\}$$

The final way to define a set is simply by listing all of its elements or listing enough of them, so that whoever is reading the definition can easily establish the pattern they follow. For example, the aforementioned set will be written as

$$S=\{1,3,5,7,9,11\}$$

Equality

Definition: Equality

Two Sets are equal iff they contain the same elements.

The Empty Set

Axiom: Existence of an Empty Set

There exists a set with no elements.

$$\exists A\forall x(x\notin A)$$

NOTATION

$$\varnothing \varnothing \{\}$$

Theorem: Uniqueness of the Empty Set

There is only one empty set.

PROOF

Let $A$ and $B$ be two empty sets. They have no elements and so the statement

$$x\in A⟺x\in B$$

is vacuously true. Therefore, $A=B$.

Theorem

The empty set is a subset of all Sets.

PROOF

Let $A$ be a set.

Suppose that the empty set $\varnothing$ is not a subset of $A$. Then there must exist an element $e\in \varnothing$ which is not an element of $A$, i.e. $e\notin A$. This is a contradiction, since the empty set contains no elements.