How to prove reflexivity, symmetry and transitivity for the following relation? [closed]
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!
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.