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# Transivity

## Basic idea

A relation is transitive if relating $a$ to $b$ and $b$ to $c$ forces $a$ related to $c$. One of the three defining properties of an equivalence relation and of a partial order.

## Key formulas

* $\forall a,b,c \in X.; (R(a,b) \land R(b,c)) \Rightarrow R(a,c)$
* Equivalently: $R \circ R \subseteq R$

A homogeneous relation **R** over set X is transitive if for all elements a,b,c in X whenever **R** relates a to b and b to c, then **R** also relates a to c.