Relation - Imported Knowlpedia Content

Reflexive: a ∼ a a\sim a a ∼ a .
Symmetric: if a ∼ b a\sim b a ∼ b , then b ∼ a b\sim a b ∼ a .
Transitive: if a ∼ b a\sim b a ∼ b and b ∼ c b\sim c b ∼ c , then a ∼ c a\sim c a ∼ c .

A relation from a set $A$ to a set $B$ is a set $R$ with

R\subseteq A\times B,

where $A\times B$ is the Cartesian product. If $(a,b)\in R$ , one often writes $a\,R\,b$ .

Remarks

A relation on a set $A$ means a relation from $A$ to itself, i.e. a subset of $A\times A$ . Special kinds of relations include equivalence relations, which encode “having the same type” in a precise sense.

Examples

The “less than or equal to” relation on $\mathbb{Z}$ is $R=\{(m,n)\in\mathbb{Z}\times\mathbb{Z}: m\le n\}$ .
For any set $A$ , the equality relation on $A$ is $\{(a,a): a\in A\}\subseteq A\times A$ .