How to determine if this relation is reflexive, symmetric, antisymmetric and transitive?

Question

We have set $X = \mathbb{C},z_1\sim z_2 \leftrightarrow$ Im $z_1$ $\leq$ Im $z_2$.

How to determine if this relation is:

• reflexive

• symmetric

• antisymmetric

• transitive?

Answer 1

Use the following definitions:

Reflexive

$z_1\sim z_1$

Symmetric

$z_1\sim z_2 \implies z_2 \sim z_1$ for $z_1\neq z_2$

Antisymmetric

$z_1\sim z_2 \text{ and }z_1\neq z_2 \implies z_2\not\sim z_1$

Transitive

$z_1\sim z_2 \text{ and } z_2\sim z_3 \implies z_1 \sim z_3$