Relation Properties

Definition. [Congruence] An integer $m \in \mathbb{Z}$ is congruent to $n$ modulo $p$ if $m - n$ is a multiple of $p \in \mathbb{Z}$. If so, we write $m \equiv n (\mod p)$ (

$m \equiv n (\mod p)$

). Example: The congruence relation $\text{Congruent}$ on some the set of integers $\mathbb{Z}$.

• It is reflexive because, for every $x \in \mathbb{Z}$, we indeed have $(x, x) \in \text{Congruent}$ since $x \equiv x (\mod p)$ because $x - x = 0$ is divisible by any integer $p$: $\frac{0}{p} = 0$.
• Since it is reflexive, it is NOT anti-reflexive.
• It is symmetric because $x \equiv y (\mod p)$ implies that $y \equiv x (\mod p)$ since, if $x - y$ is a multiple of $p$, then $y - x = -(x - y)$ is a multiple of $p$, too.
• It is NOT anti-symmetric because, if $x \equiv y (\mod p)$ and $y \equiv x (\mod p)$, it doesn't necessarily mean that $x = y$. For example, if $p = 2$, then $4 \equiv 8 (\mod 2)$ and $8 \equiv 4 (\mod 2)$, but $4 \neq 8$.
• It is transitive since $x \equiv y (\mod p)$ and $y \equiv z (\mod p)$ implies that $x \equiv z (\mod p)$. This is because $x - z = (x - y) + (y - z)$ is clearly a multiple of $p$.

In summary, the $\text{Congruent}$ relation has the following $3$ properties: reflexive, symmetric, and transitive.