Set is Subset of Itself

Theorem

Every set is a subset of itself:

$\forall S: S \subseteq S$


Thus, by definition, the relation is a subset of is reflexive.


Proof

\(\ds \forall x: \, \) \(\ds \leftparen {x \in S}\) \(\implies\) \(\ds \rightparen {x \in S}\) Law of Identity: \(\quad\) a statement implies itself
\(\ds \leadsto \ \ \) \(\ds S\) \(\subseteq\) \(\ds S\) Definition of Subset

$\blacksquare$


Sources

  • 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 1$: The Axiom of Extension
  • 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
  • 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$
  • 1964: William K. Smith: Limits and Continuity ... (previous) ... (next): $\S 2.1$: Sets
  • 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): $\S 1.2$. Subsets
  • 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
  • 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.2$. Sets
  • 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B i}$
  • 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.3$: Subsets
  • 1971: Patrick J. Murphy and Albert F. Kempf: The New Mathematics Made Simple (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets: Subsets
  • 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems: Theorem $1.1 \ \text{(a)}$
  • 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets
  • 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6.1$: Subsets
  • 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets
  • 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Sets: Definition $3$
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subset (i)
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