Can anyone offer guidance on how to determine the following relation as being reflexive, symmetric or antisymmetric when the relation is given by
$$ x \sim y \iff (x - y) > 0 \land (x - y) < 4. $$
I think it is not reflexive and antisymmetric.
Any assistance is appreciated, thanks.
Recall that a relation, $\sim$, on some set $A$ is reflexive if, and only if,
$$x\sim x \quad \forall x\in A$$
and is antisymmetric if, and only if,
$$\forall x,y\in A, \quad x\sim y \quad \land \quad y\sim x \implies x=y$$
Showing that your relation is not reflexive should be straight forward (is $x-x > 0$?).
Showing antisymmetry for your relation would be difficult (since it isn't) and to show this, can you find any $x,y \in \mathbb{R}$ such that $x-y >0$, $x-y < 4$ and $x\neq y$?
Additionally, is it symmetric? This requires that if $x\sim y$ then $y\sim x$. That is, can you show that for every $x,y \in \mathbb{R}$ where $x-y >0$ and $x-y < 4$, is $y-x >0$ and $y-x <4$?
