Defining a relation to a set

Question

I have a homework question that asks me to define a relation A2 on $Z$ which is an equivalence relation containing three equivalence classes.

$$Z = \{a, b, c, d, e\}$$

I understand what equivalence classes are, but does defining a relation on set $Z$ mean that I'm creating another set to multiply set $Z$ and then make sure the result is reflexive, symmetric etc. ?

Answer 1

Defining a relation on set $Z$ means relation in $Z\times Z$.
So an example is $\{(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),(c,c),(d,d),(e,e)\}$

Answer 2

Let me give you an example. Define relation $\sim$ on $Z$ so that the following seven relations hold:

$$a \sim b,\hspace{5mm} b\sim a$$ $$a \sim a, \hspace{5mm} b \sim b,\hspace{5mm} c \sim c,\hspace{5mm} d \sim d,\hspace{5mm} e \sim e$$ while any other relation is false (like $e \sim a$).

You can check that $\sim$ is reflexive, symmetric, and transitive to convince yourself that this is really an equivalence relation.

Under this equivalence relation, $a$ and $b$ are the same but every other pair of distinct elements is different. The equivalence classes are therefore: $$\{a,b\}, \{c\}, \{d\}, \{e\}$$ and there are 4 of them. Can you see how to define a new relation with only 3 equivalence classes?

Answer 3

Split $Z$ up in $3$ disjoint and nonempty subsets $Z_1,Z_2,Z_3$ with $Z=Z_1\cup Z_2\cup Z_3$.

Now define a relation $R\subset Z\times Z$ by stating that: $$uRv\iff\{u,v\}\subseteq Z_1\vee\{u,v\}\subseteq Z_2\vee\{u,v\}\subseteq Z_3$$

Then $R$ is an equivalence relation and $Z_1,Z_2,Z_3$ are its equivalence classes.