# Definition:Set/Uniqueness of Elements

## Definition

A set is **uniquely determined** by its elements.

This means that the only thing that defines **what a set is** is **what it contains**.

So, how you choose to **list** or **define** the contents makes **no difference** to what the contents actually **are**.

### Multiple Specification

For a given **set**, an object is either **in** the set or **not in** the set.

So, if an element is in a set, then it is in the set **only once**, however many times it may appear in the definition of the set.

Thus, the set $\set {1, 2, 2, 3, 3, 4}$ is the same set as $\set {1, 2, 3, 4}$.

$2$ and $3$ are in the set, and **listing** them twice makes no difference to the set's **contents**.

Like the membership of a club, if you're in, you're in -- however many membership cards you have to prove it.

### Order of Listing

It makes no difference in what order the elements of a set are specified.

This means that the sets $S = \set {1, 2, 3, 4}$ and $T = \set {3, 4, 2, 1}$ are the *same set*.

### Equality of Sets

By definition of **set equality**

$S$ and $T$ are equal if and only if they have the same elements:

- $S = T \iff \paren {\forall x: x \in S \iff x \in T}$

So, to take the club membership analogy, if two clubs had exactly the same members, the clubs would be considered as *the same club*, although they may be given different **names**.

This follows from the definition of **equals** given above.

Note that there **are** mathematical constructs which **do** take into account both (or either of) the order in which the elements appear, and the number of times they appear, but these are *not* **sets** as such.

## Also see

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): Chapter $1$: A Common Language: $\S 1.1$ Sets - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 2$ - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set? - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions