If $A = \{1,2,3\}$, $R$ an equivalence relation on $A$ if $R = \{(1,1), (2,2), (3,3)\}$?
I'm having trouble understanding when a relation is symmetric, antisymmetric, or transitive. Does the symmetric property require that ordered pairs $(a,b)$ and $(b,a)$ be present in $R$?
The three properties required for an equivalence relation are 1) reflexive: for any $x$ in the set $(x, x)$ is in the relation. That is obviously true here- the set is $\{1, 2, 3\}$ and we have $(1, 1), (2, 2), (3, 3)$ in the relation.
2) symmetric: if $(x, y)$ is in the relation then so is $(y,x)$. Again that is obvious. The only pairs in the relation are $(1, 1), (2, 2)$, and $(3, 3)$. Reversing the order just gives the same thing again.
3) transitive: if $(x, y)$ and $(y, z)$ are in the relation, then so is $(x, z)$. Once again, obvious! $(x, y)$ is in the relation only if $x= y$ but then we must have $y= z$ so $x= z$. That is the same as $(z, z)$.
Actually, while there can be very complicated "equivalence relation", this particular one is the "epitome" of equivalence relations- it is the identity relation. "$x$ is equivalent to $y$" if and only if "$x$ is equal to $y$".
(Anti-symmetric plays no role in equivalence relations but since you ask: where "symmetric" requires "if $(x, y)$ is in the relation, then $(y, x)$ is also". "Anti-symmetric" requires "if $(x, y)$ is in the relation, then $(y, x)$ is not.)