Subset Relation is Transitive
Jump to navigation
Jump to search
Theorem
The subset relation is transitive:
- $\paren {R \subseteq S} \land \paren {S \subseteq T} \implies R \subseteq T$
Proof 1
\(\ds \) | \(\) | \(\ds \paren {R \subseteq S} \land \paren {S \subseteq T}\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {x \in R \implies x \in S} \land \paren {x \in S \implies x \in T}\) | Definition of Subset | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \paren {x \in R \implies x \in T}\) | Hypothetical Syllogism | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds R \subseteq T\) | Definition of Subset |
$\blacksquare$
Proof 2
Let $V$ be a basic universe.
By definition of basic universe, $R$, $S$ and $T$ are all elements of $V$.
By the Axiom of Transitivity, $R$, $S$ and $T$ are all classes.
We are given that $R \subseteq S$ and $S \subseteq T$.
Hence by Subclass of Subclass is Subclass, $R$ is a subclass of $T$.
By Subclass of Set is Set, it follows that $R$ is a subset of $T$.
Hence the result.
$\blacksquare$
Sources
- 1915: Georg Cantor: Contributions to the Founding of the Theory of Transfinite Numbers ... (previous) ... (next): First Article: $\S 1$: The Conception of Power or Cardinal Number
- 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): $\S 1$: The Axiom of Extension
- 1963: George F. Simmons: Introduction to Topology and Modern Analysis ... (previous) ... (next): $\S 1$: Sets and Set Inclusion
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Exercise $\text{B iii}$
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets: Exercise $2$
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets: Problem Set $\text{A}.1$: $5$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 6.2$: Subsets
- 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Theorem $\text{A}.6$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): subset (iii)