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

Proper supersets