7 Equivalence Relations and Partitions

7.1 Introducing Relations

7.1.1 A More General “Function”

Example 7.1.1.

For each function given below, either confirm that the rule satisfies the definition of a function, or give an example of where the rule fails the definition of a function. You may want to fill in the function diagrams to help with your explanation.

(a)

$f:\mathbb{Z}\rightarrow\mathbb{Z}/5\mathbb{Z}$ , $f(x)=[x]_{5}$

.
(b)

$g:\mathbb{Z}/5\mathbb{Z}\rightarrow\mathbb{Z}$ , $g([x]_{5})=x$

.

We review a definition we gave back a few sections ago:

Definition 7.1.2.

Let $A$ and $B$ be sets. A relation $R$ from $A$ to $B$ is a subset of $A\times B$ .

We did not discuss this very much when initially defining it; it was mere scaffolding, not an object of study in itself. This is for good reason: relations are harder to visualize from their definition.

The best visualization that can be offered comes in the form of an example: a relation from $\mathbb{R}$ to $\mathbb{R}$ is a generalization of a function in that it does not have to pass the vertical line test. The function $x=y^{2}$ is a perfect example:

However, relations are meant to be much, much more general than this. Part (b) of our first example in this section gives another example of what a relation can be: any one input can yield infinitely many outputs.

Here is a third example: let

R=\{(x,y)\in\mathbb{R}\times\mathbb{R}\mid x<y\}.

Then $(2,3)\in R$ while $(3,-3)\notin R$ . We will sometimes use the notation “ $2\,R\,3$ ” and “ $3\not R-3$ ” to describe these statements. This relation $R$ has a better name: “ $<$ ”, which is the one typically used. So we would write $2<3$ and $3\not<-3$ .

7.1.2 Proving with Relations

Example 7.1.3.

Let $\mathbb{Z}^{*}:=\mathbb{Z}\setminus\{0\}$ . Define the relation $\sim$ (LaTeX $\backslash$ sim) on $\mathbb{Z}\times\mathbb{Z}^{*}$ by, for all $a,c\in\mathbb{Z}$ and all $b,d\in\mathbb{Z}^{*}$ ,

(a,b)\sim(c,d)\quad\text{iff}\quad ad=bc.

Give a few pairs of points in $\mathbb{Z}\times\mathbb{Z}^{*}$ that are related. Do you notice any patterns? Make a conjecture of what our relation signifies.

Now that we have shown how general a relation is, let us show the function as a special type of relation.

Definition 7.1.4.

Let $R$ be a relation from a set $X$ to a set $Y$ ; i.e., $R\subseteq X\times Y$ . The domain of $R$ is the set

\operatorname{dom}R:=\{a\in X\mid(\exists b\in Y)a\,R\,b\}.

A relation $R$ from a set $X$ to a set $Y$ is a function if for all $a\in\operatorname{dom}R$ there exists a unique $b\in Y$ such that $a\,R\,b$ . I.e.,

(\forall a\in\operatorname{dom}R)(\exists!b\in Y)a\,R\,b.

This definition is vague enough to never see much use. We only include it as a formality. However, you might notice several connections between this definition and the definition of a function given in Section 5.1.