Relation defined on $\mathbb{R}$ Started studying relations and attempted a very basic problem which I think I have done correctly, can somebody tell me if there are any flaws?Also if a relation is reflexive or transitive can it have trichotomy?I know that if it is symmetric it cannot.
Consider the relation on $\mathbb{R}$ defined by $n \simeq m$ iff $n-m \in \mathbb{Z}$
Is the relation Reflexive,symmetric,and transitive?
Attempt:
$n-n=0$ and $0 \in \mathbb{Z}$ So reflexive
Suppose $n-m \in \mathbb{Z}$ Then $-(n-m) \in \mathbb{Z}$ So symmetric
Suppose $n-m \in \mathbb{Z}$and $m-p  \in \mathbb{Z}$
then $(n-m)+(m-p) \in   \mathbb{Z}$ so transitive.
Does the relation of trichotomy?
Relation does not have trichotomy since the relation is symmetric so
$nRm$ and $mRn$ are true which contradicts trichotomy
 A: When showing that a relation does not have a property, usually the fastest and most obvious way is to give a concrete counterexample. For instance, $1\simeq0$ and $0\simeq1$, or $0\not\simeq\frac12, \frac12\not\simeq0$ and $0\neq\frac12$, either pair proving that the relation doesn't have trichotomy.
Of course, some times general arguments can show impossibilities as well. Such as no relation (on a set with more than one element) being both reflexive and having trichotomy. This is because for a relation $R$ with trichotomy, and any two elements $a,b$, exactly one of $aRb, bRa$ or $a=b$ holds. But for two unequal elements, this breaks reflexivity.
As for the rest, it is correct, but I personally feel it ought to be more explicit: actually spell out exactly what makes the property hold. For symmetry, I would write something along the lines of

Assume $m\simeq n$. This means $m-n\in\Bbb Z$. But then $n-m=-(m-n)\in\Bbb Z$ as well, so we get $n\simeq m$, showing that the relation is symmetric.

and for transitivity

Assume $m\simeq n$ and $n\simeq p$. Then $m-n\in\Bbb Z$ and $n-p\in\Bbb Z$. But then $m-p=(m-n)+(n-p)\in\Bbb Z$ as well, showing that the relation is transitive.

In other words, expand a bit more on what exactly it is your doing. Not just write down the central equality of your argument and let the reader full in the rest.
