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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?
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 $(2,4)\not\in S$.
None of the two relations is antisymmetric. Apart from that I do agree.
