Definition:Set/Distinction between Element and Set

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
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): Appendix $\text{A}$: Set Theory: Unordered Pairs and their Relatives
• 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 $1$.