# Relations

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## Definitions: Divisibility and Relations


**Definition:** Let $n$ and $k$ be positive integer. We say that $k$ divides $n$ if there exists an integer $a$ such that $n = ak$.

**Definition:** A *relation* on a set $S$ is a subset of $S \times S$.

**Example:** We can be **divides** as a relation on the integers: let $D$ be the set
$$\{ (a,b) \mid a,b \in \mathbb{Z} \text { and } a \text{ divides } b \}.$$

<strong> Example: </strong> We can also view *less than or equal to* as a relation, the set
$$\{ (a,b) \mid a,b \in \mathbb{Z} \text{ and } a \leq b \}$$.

<strong> Another Example: </strong> The *Penland relation* can be defined as $\{(a,b) \mid a,b \in \mathbb{Z}, a+b = 7 \}$.

## More Examples



## Properties a Relation on a Set Can Have

**Definition:** Let $R$ be a relation on a set $A$. We say that $R$ is *reflexive* if $(a,a) \in R$ for any $a \in A$.

**Definition:** Let $R$ be a relation on a set $A$. We say that $R$ is *symmetric* if whenever $(a,b) \in R$, we also have that $(b,a) \in R$.

**Definition:** Let $R$ be a relation on a set $A$. We say that $R$ is *symmetric* if whenever $(a,b) \in R$ and $(b,c) \in R$, it also follows that $(a,c) \in R$.

Here is a nice video that explains these three properties:

<iframe width="480" height="280" src="https://www.youtube.com/embed/q0xN_N7l_Kw" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>