Let R be binary relation on N (natural numbers) defined by xRy if and only if x − 2 ≤ y ≤ x + 2. Is R reflexive? Is R symmetric? Is R antisymmetric? Is R transitive?
I'm not sure if I'm understanding this right or how I can prove it. This is what I currently have:
Reflexive: Yes, because $x-2≤x≤x+2$ so $xRx$ (How can I prove this for all $x$ though?)
Symmetric: Yes, because $xR(x+2)$ therefore $(x+2)R((x+2)-2)$ (I can tell this is wrong, but basically I'm trying to say since x always relates to x+2, x+2 will always relate to x).
Anti-Symmetry and Transitive just need examples of them not working, so I think these work:
Anti-Symmetric: No, e.g. $2R4$ and $4R2$, but $4 ≠ 2$
Transitive: No, e.g. $2R4$ and $4R6$, but $2! R 6$
Thanks for any help.