Binary relation

In mathematics, a binary relation associates some elements of one set called the domain with some elements of another set (possibly the same) called the codomain. Precisely, a binary relation over sets X and Y is a set of ordered pairs (x,y), where x is an element of X and y is an element of Y. It encodes the common concept of relation: an element x {\displaystyle x} is related to an element y {\displaystyle y} if and only if the pair ( x , y ) {\displaystyle (x,y)} belongs to the set of ordered pairs that defines the binary relation. An example of a binary relation is the "divides" relation over the set of prime numbers P {\displaystyle \mathbb {P} } and the set of integers Z {\displaystyle \mathbb {Z} } , in which each prime p {\displaystyle p} is related to each integer z {\displaystyle z} that is a multiple of p {\displaystyle p} , but not to an integer that is not a multiple of p {\displaystyle p} . In this relation, for instance, the prime number 2 {\displaystyle 2} is related to numbers such as − 4 {\displaystyle -4} , 0 {\displaystyle 0} , 6 {\displaystyle 6} , 10 {\displaystyle 10} , but not to 1 {\displaystyle 1} or 9 {\displaystyle 9} , just as the prime number 3 {\displaystyle 3} is related to 0 {\displaystyle 0} , 6 {\displaystyle 6} , and 9 {\displaystyle 9} , but not to 4 {\displaystyle 4} or 13 {\displaystyle 13} . A binary relation is called a homogeneous relation when X = Y {\displaystyle X=Y} . A binary relation is also called a heterogeneous relation when it is not necessary that X = Y {\displaystyle X=Y} . Binary relations, and especially homogeneous relations, are used in many branches of...