Mathematical Reasoning: Writing and Proof (PreTeXt Edition)

Beginning Activity Beginning Activity 1: Properties of Relations

A4US

In previous mathematics courses, we have worked with the equality relation. For example, let $R$ be the relation on $\mathbb{Z}$ defined as follows: For all $a,b\in \mathbb{Z}\text{,}$ $aRb$ if and only if $a=b\text{.}$ We know this equality relation on $\mathbb{Z}$ has the following properties:

• For each $a\in \mathbb{Z}\text{,}$ $a=a$ and so $aRa\text{.}$

• For all $a,b\in \mathbb{Z}\text{,}$ if $a=b\text{,}$ then $b=a\text{.}$ That is, if $aRb\text{,}$ then $bRa\text{.}$

• For all $a,b,c\in \mathbb{Z}\text{,}$ if $a=b$ and $b=c\text{,}$ then $a=c\text{.}$ That is, if $aRb$ and $bRc\text{,}$ then $aRc\text{.}$

In mathematics, when something satisfies certain properties, we often ask if other things satisfy the same properties. Before investigating this, we will give names to these properties.

Before exploring examples, for each of these properties, it is a good idea to understand what it means to say that a relation does not satisfy the property. So let $A$ be a nonempty set and let $R$ be a relation on $A\text{.}$

To illustrate these properties, we let $A=\left\{1,2,3,4\right\}$ and define the relations $R$ and $T$ on $A$ as follows: