How to prove reflexivity, symmetry and transitivity for the following relation?

Question

I would like to know how to prove reflexivity, symmetry and transitivity for $\sim$ according to the following definition:

Suppose $\sim$ is defined on the set of the integers as follows : $a\sim b$ iff $ab ≤ a|b|$

Please help me. Thanks!

Answer 1

If $-1 \sim -1$ then $1 \le -1$.

In other words, the relation is not reflexive.

Answer 2

In general, you would prove it by showing:

1) Reflexive: $a \sim a$.

2) Symmetric: If $a\sim b$, then $b\sim a$.

3) Transitive: If $a\sim b$ and $b\sim c$, then $a\sim c$.

However, you're going to have trouble, because it's not true for your relation. Note that $-1 \sim -1$ is false, because $-1\cdot -1 \leq -1\lvert -1 \rvert$ is false.