# Determine If relations are reflexive, symmetric, antisymmetric, transitive

For each of the following relations on the set $\mathbb{Z}$ of integers, determine if it is reflexive, symmetric, anti-symmetric, or transitive. On the basis of these properties, state whether or not it is an equivalence relation or a partial order.

(a) $R = \{(a, b) \in \mathbb{Z^2} : a^2 = b^2\}$.

(b) $S = \{(a, b) \in \mathbb{Z^2} : \mid a − b \mid \le 1\}$.

Am I right to say that (a) is reflexive, symmetric, antisymmetric, and transitive?

And (b) is reflexive, symmetric, and antisymmetric?

• a) -x R x and x R -x but x != -x so not antisymmetric. In general anti-symmetric and symmetric are opposites and the only way a relation can be both is if no unequal items are ever related. Nov 29, 2016 at 21:38

In my opinion the first relation (a) is indeed reflexive, symmetric and transitive but not antisymmetric, as $(-2,2) \in R$ and $(2,-2) \in R$, but $2\neq -2$.
The second relation (b) is indeed reflexive and symmetric, but again not antisymmetric as $(0,1)\in S$ and $(1,0)\in S$, but $1\neq 0$. Transitivity also fails: Take $(2,3) \in S$ and $(3,4)\in S$, then obviously $(2,4)\not\in S$.