# Definition:Set/Uniqueness of Elements

## Contents

## 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 also makes no difference what order the elements 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

Two sets which have exactly the same elements are the same, whatever the sets are called.

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): $\S 1.1$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 2$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S1.1$: Sets and Membership - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets - 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? - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $5$ - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.1$: Definition $\text{A}.1$