# Integer equal to

## Definition

$$(a, b) \sim (c, d) \Leftrightarrow a + d = b + c$$
THEOREM

The equal to relation $$\sim$$ is reflexive on the set of integers $$\mathbb{Z}$$.

$$\forall x \in \mathbb{Z} : x \sim x$$

Proof available

THEOREM

The equal to relation $$\sim$$ is symmetric on the set of integers $$\mathbb{Z}$$.

$$\forall x,y \in \mathbb{Z} : x \sim y \Rightarrow y \sim x$$

Proof available

THEOREM

The equal to relation $$\sim$$ is transitive on the set of integers $$\mathbb{Z}$$.

$$\forall x,y,z \in \mathbb{Z} : x \sim y \wedge y \sim z \Rightarrow x \sim z$$

Proof available