
# 2: Binary Relations

Definition

Let $$S$$ be a non-empty set. A binary relation $$R$$ on $$S$$ is a rule that relates two elements in $$S$$.

In terms of set,  $$R \subset S\times S.$$

Example $$\PageIndex{1}$$:

Let $$S=\{1,2,3\}$$, and let $$R$$ be the relation $$<$$.

Then

1. $$1$$ is related to $$2$$., we denote this by $$1 R 2$$ or $$1 < 2$$.
2. $$2$$ is not related to $$1$$., we denote this by $$2 \not{R} 1$$ or $$2 \nless$$.1

#### Example $$\PageIndex{2}$$:

The following are binary relations on \mathbb Z.

1. $$=$$
2. $$<$$
3. $$>$$
4. $$\leq$$
5. $$geq$$

Definition

Let $$a$$ and $$d$$ be integers. The $$d$$ divides $$a$$, denoted by $$d|a$$, if there exists an integer $$k$$ such that $$a=dk.$$

In other words,

•  $$a$$ is multiples of $$d$$
• $$d$$ is a factor of $$a$$
• $$a$$ is divisible by $$d$$

Example $$\PageIndex{3}$$:

1. $$4|2$$
2. $$2|4$$

Example $$\PageIndex{4}$$:

Define  a relation $$R$$ on $$S={\mathbb Z}$$ by:
$$a \,R\, b$$ if and only if $$2 \mid a-b.$$

### Properties

Definition: Reflexive

Let $$R$$ be relation on a set $$S$$, then $$R$$ is said to be reflexive if  $$a R a, \forall a \in S$$.

Example $$\PageIndex{6}$$:

Define  a relation $$R$$ on $$S={\mathbb Z}$$ by:
$$a \,R\, b$$ if and only if $$2 \mid a-b.$$

Is $$R$$ Reflexive?