Transitivity

Transitivity in the Context of Relations

Definition: In logic and mathematics, a relation $R$ on a set $S$ is said to be transitive if, whenever the relation holds between a pair of elements in the set, and also between the second of these elements and a third element, then it necessarily holds between the first and the third element as well. Formally, for a relation $R$ on a set $S$, $R$ is transitive if for all $a, b, c \in S$, if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

Mathematical Formulation:

1. Set and Relation: Consider a set $S$ and a relation $R \subseteq S \times S$, where $S \times S$ denotes the Cartesian product of $S$ with itself.
2. Transitive Property: The relation $R$ is transitive if, the conditions $(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R \forall a, b, c \in S$.