# Definition:Set/Distinction between Element and Set

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## Definition

It is important to distinguish between an element, for example $a$, and a singleton containing it, that is, $\set a$.

That is $a$ and $\set a$ are *not* the same thing.

While it is true that:

- $a \in \set a$

it is not true that:

- $a = \set a$

neither is it true that:

- $a \in a$

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 3$: Unordered Pairs - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{B} \ 5$ - 1965: A.M. Arthurs:
*Probability Theory*... (previous) ... (next): Chapter $1$: Set Theory: $1.2$: Sets and subsets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.1$. Sets: Example $5$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): Chapter $1$: Sets, Functions, and Relations: $\S 1$: Sets and Membership - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 4$ The pairing axiom: Note $1$.

- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Unordered Pairs and their Relatives