Definition:Transitive Relation

Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

$\mathcal R$ is transitive iff:

$\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \in \mathcal R$

Also see

• Transitivity

• Antitransitive Relation
• Non-transitive Relation

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 7$: Relations
• W.E. Deskins: Abstract Algebra (1964): $\S 1.2$: Definition $1.5$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 2.2$: Definition $3$
• Seth Warner: Modern Algebra (1965): $\S 10$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.3$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$ Theorem $3$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.4$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 1.6$: Theorem $3 \ \text{(iii)}$
• George E. Andrews: Number Theory (1971): $\S 4.1$: Theorem $4.1$ (Remark)
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 16 \ \text{(c)}$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$
• John F. Humphreys: A Course in Group Theory (1996): $\S 2$: Definition $2.24 \ (3)$