Definition:Set/Uniqueness of Elements/Equality of Sets

Definition

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.

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets