A binary relation \( R \) is a \( 2 \)-ary relation, when \( \left ( x , y \right ) \in R \), then we write \( x R y \) to say that \( x \) is related to \( y \).
Note: A binary relation on \( X \times X \) is stated simply as a relation on \( X \)
An equivalence relation is a reflexive, symmetric and transitive relation
when \( R \) is an equivalence relation, we denote it by \( \sim \) instead of \( R \)
Given any \( a \in G \), then clearly \( a \% n = a \% n \) so that \( a R a \), if \( a R b \), then \( a \% n = b \% n \) and then \( b \% n = a \% n \), finally if \( a \% n = b \% n \) and \( b \% n = c \% n \) therefore \( a \% n = c \% n \)
A partial order is a binary relation denoted by \( \le \) on a set \( X \) such that \( \le \) is reflexive, anti-symmetric and transitive relation.
With the strict partial order \( \subset \) we can see that \( \left\{ 1 \right\} \) and \( \left\{ 2 \right\} \) are incomparable.