Mathematics

This problem asks you to recognize that $R$ is a congruence relation — a way of grouping integers that behave the same way modulo 3. The key insight is that $R$ partitions $\mathbb{Z}$ into three equivalence classes (remainders 0, 1, and 2 when divided by 3), and you should check whether $R$ is reflexive, symmetric, transitive, or some combination. Think about what it means for $(a,b) \in R$: does every integer relate to itself? If $a$ relates to $b$, does $b$ relate to $a$? If $a$ relates to $b$ and $b$ relates to $c$, does $a$ relate to $c$?

This problem asks you to recognize that $R$ is a congruence relation — a way of grouping integers that behave the same way modulo 3. The key insight is that $R$ partitions $\mathbb{Z}$ into three equivalence classes (remainders 0, 1, and 2 when divided by 3), and you should check whether $R$ is reflexive, symmetric, transitive, or some combination. Think about what it means for $(a,b) \in R$: does every integer relate to itself? If $a$ relates to $b$, does $b$ relate to $a$? If $a$ relates to $b$ and $b$ relates to $c$, does $a$ relate to $c$?