Definition:Set/Uniqueness of Elements

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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

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