Is my understanding of anti-symmetry and symmetry correct?

Solution 1:

First, there should be $\forall $ statements (e.g. reflexive is $\forall a$, $(a,a)\in R$ and transitive is $\forall a,b,c$, $(a,b)\in R\land (b,c)\in R\implies (a,c)\in R$).

Secondly, for your definition of antisymmetric, $a$ and $b$ must be distinct. We can have e.g. $(1,1)$ in an antisymmetric relation.

So your example of a symmetric relation is correct and your example of an anti-symmetric relation would be correct even if you added $(2,2)$ to the relation.

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