Consider the relation $R$ on $\mathbb Z$ as: $\forall m,n\in \mathbb{Z}, mRn \iff m−n \text{ is odd}$ . Is $R$ reflexive, symmetric, or transitive?

Question

Consider the relation $R$ on $\mathbb Z$ as: $\forall m,n\in \mathbb{Z}, mRn \iff m−n \text{ is odd}$. Is $R$ reflexive, symmetric, or transitive?

Provide a complete proof or counterexample for each property. And I have no idea what I'm doing.

Answer 1

Well, if $m R n$, then $m - n$ is odd for $m,n \in \mathbb{Z}$.

So, let's check all the properties...

Reflexive: $m R m$ iff $m-m$ is odd but $m-m = 0$ so reflexivity doesn't hold. So, pick your favorite integer for a counterexample (eg. take $m=3$ and we see $3-3 = 0$)

Symmetric: $m R n$ iff $m-n = 2k+1$ where $k \in \mathbb{Z}$. Note that if $2k +1$ is odd, $-2k-1 = n - m$ is odd as well so $n R m$.

Transitivity: Let $l,m,n \in \mathbb{Z}$. Then we want to show if $m R n$ and $n R l$, then $m R l$. That is, if $m-n$ is odd and $n-l$ is odd, we want to show $m-l$ is odd. Note that "odd" plus "odd" is even so $$(m-n) + (n-l) = m -l$$ is even so transitivity doesn't hold. A counterexample could be $m=4, n =3,l = 2$. Then $m R n$ since $4-3 = 1$ is odd and $n R l$ since $3-2 = 1$ is odd but $m R l$ is false since $m-l = 4-2 = 2$ is even. Try coming up with another example.

Here's another exercise for you to try. Let $m R n$ iff $n >m$. Check for reflexivity, symmetry, and transitivity (get used to working with definitions!).