Reflexive, irreflexive, symmetric, asymmetric, antisymmetric or transitive? Let S be the relation on the set of integers and xSy is defined as x and y yield the same remainder when divided by 3. Determine whether the relation is reflexive, irreflexive, symmetric, asymmetric, antisymmetric or transitive. Give a counterexample if your answer is "No". How to know that the number of x and y? Sorry, i have no idea to start with this kind of question.

The relation is nothing but $xRy \iff x\equiv y(mod 3)$ i.e $x Ry \iff x-y$ is divisible by $3$..which is an equivalence relation

Definitely not antisymmetric $8 R 14$ and $14 R 8$ but $8\
eq 14$