Problem Statement
Show that the relation $R$ in the set $\mathbb{Z}$ of integers given by $R = \{(a, b) : 2 \text{ divides } (a - b)\}$ is an equivalence relation.
Verified Solution & Marking Scheme
Reflexive Property
For any $a \in \mathbb{Z}$, $a - a = 0 = 2(0)$. Since 2 divides 0, $(a, a) \in R$. Hence $R$ is reflexive.
Symmetric Property
Let $(a, b) \in R$. Then $a - b = 2k$ for $k \in \mathbb{Z}$. Then $b - a = -2k = 2(-k)$. Since $-k \in \mathbb{Z}$, $(b, a) \in R$. Hence $R$ is symmetric.
Transitive Property
Let $(a, b) \in R$ and $(b, c) \in R$. Then $a - b = 2k$ and $b - c = 2m$. Summing gives $a - c = (a - b) + (b - c) = 2(k + m)$. Since $k + m \in \mathbb{Z}$, $(a, c) \in R$. Hence $R$ is transitive.