Question on equivalence relations

Question

In set theory a relation is said to be equivalence if the relation is,.

1. Reflexive
2. Symmetric
3. Transitive

I would like to know if the following relation is an equivalence one.

$R = \{ (m,n) \in Z \times Z \; : \; m - n \text{ is even} \}$,

where $Z$ is the set of all integers.

Here's why I think the relation is not equivalence.

1. To be reflexive let us consider $m = n = 4$. By the relation $m-n = 0$, which is not even. Thus it is not reflexive.

... But the text book I'm referring to (Analysis by S.R Lay) states that the relation is an equivalence one.

Can someone explain to me why it is such. Thanks in advance. :)

Answer 1

1. $m-m=0$ is even. So reflexivity holds.

2. Suppose $m-n=2k$ (i.e. that $m-n$ is even). Then $n-m=-2k$, so n-m is also even. So the symmetric property holds.

3. Suppose $m-n=2k$, and $l-m=2j$. Then $l-n=l-m+m-n=2k+2j=2(k+j)$, so $m-n$ and $l-m$ even implies $l-n$ even. So transitivity holds.