Question

Define the relation R as
R = {(a, b)|a, b ∈ Z, 4 divides a − b}. Show that R is reflexive, transitive and symmetric.

Answer 1

Answer on Question #76364 – Math – Discrete Mathematics

Question

Define the relation $R$ as

$R = \{(a, b) \mid a, b \in \mathbb{Z}, 4 \text{ divides } a - b\}$. Show that $R$ is reflexive, transitive and symmetric.

Solution

1. Reflexive property: $a - a = 0 \mid 4$ for every $a \in \mathbb{Z}$.

2. Transitive property: another definition for this relation is $R = \{(a, b) \mid a, b \in \mathbb{Z}, a \equiv b \pmod{4}\}$. As congruence relation is transitive $R$ is transitive.

3. Symmetric property: if 4 divides $a - b$, then 4 divides $b - a$, so $R$ is symmetric.

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