Definition:Set/Uniqueness of Elements/Multiple Specification
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Definition
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.
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
- 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Sets
- 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
- 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 $\text{A}.1$