Determine whether the relation R on the set A is reflexive, irreflexive, symmetric, asymmetric, anti-symmetric, or transitive.

Question

Determine whether the relation R on the set A is reflexive, irreflexive, symmetric, asymmetric, anti-symmetric, or transitive. (a) A=Z; a R b if and only if a≤ b + 1 (b) A=Z+; a R b if and only if |a − b| ≤ 2 (c) A=Z+ ; a R b if and only if a = b^k for some k ∈ Z+ (d) A=Z; a R b if and only if a + b is even. (e) A=Z; a R b if and only if |a − b| = 2

Answer 1

We proceed as follows:
a)$\alpha\lt{\alpha+1}$, thus R is reflexive. If $\alpha\leq\beta+1$, then obviously if $\beta\leq\alpha+1$,then $\alpha=\beta$, since $\alpha-1\leq\beta\leq\alpha+1$ and $\beta\in\mathbb{Z}$.Since the relation only holds in a special case, the relation is anti-symmetric.
b)${aRa}$, since ${|a-a|=0\leq2}$, symmetry follows form the evenness of the absolute value function $(|a-b|=|b-a|=|-(a-b)|)$
c)From b) we infer that the relation is irreflexive, yet symmetrical.