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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?

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    $\begingroup$ 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. $\endgroup$
    – fleablood
    Nov 29, 2016 at 21:38

2 Answers 2

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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 $(2,4)\not\in S$.

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  • $\begingroup$ Can't believe I didn't see that, Thanks! $\endgroup$
    – Jake0991
    Nov 29, 2016 at 21:43
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None of the two relations is antisymmetric. Apart from that I do agree.

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