Divides in the context of Set-like relation

In mathematics, the greatest common divisor (GCD), also known as greatest common factor (GCF), of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers x, y, the greatest common divisor of x and y is denoted ${\displaystyle \gcd(x,y)}$. For example, the GCD of 8 and 12 is 4, that is, gcd(8, 12) = 4.

In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor, etc. Historically, other names for the same concept have included greatest common measure.

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 ${\displaystyle X}$ and ${\displaystyle Y}$ is a set of ordered pairs ${\displaystyle (x,y)}$, where ${\displaystyle x}$ is an element of ${\displaystyle X}$ and ${\displaystyle y}$ is an element of ${\displaystyle Y}$. It encodes the common concept of relation: an element ${\displaystyle x}$ is related to an element ${\displaystyle y}$, if and only if the pair ${\displaystyle (x,y)}$ 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 ${\displaystyle \mathbb {P} }$ and the set of integers ${\displaystyle \mathbb {Z} }$, in which each prime ${\displaystyle p}$ is related to each integer ${\displaystyle z}$ that is a multiple of ${\displaystyle p}$, but not to an integer that is not a multiple of ${\displaystyle p}$. In this relation, for instance, the prime number ${\displaystyle 2}$ is related to numbers such as ${\displaystyle -4}$, ${\displaystyle 0}$, ${\displaystyle 6}$, ${\displaystyle 10}$, but not to ${\displaystyle 1}$ or ${\displaystyle 9}$, just as the prime number ${\displaystyle 3}$ is related to ${\displaystyle 0}$, ${\displaystyle 6}$, and ${\displaystyle 9}$, but not to ${\displaystyle 4}$ or ${\displaystyle 13}$.

In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true. They are called incomparable if they are not comparable.