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.

Siblings

• AntiSymmetry
• Reflexivity