Finding the equivalence class of a relation |a| = |b|

Question

For the following relation $R$ on the set $X$ determine whether it is $(i)$ reflexive, $(ii)$ symmetric and $(iii)$ transitive. Give proofs or counter examples. In the case where $R$ is an equivalence relation, describe the equivalence classes of $R$.

$X=\mathbb C,\; a\;R\;b\iff |a|=|b|$

$(i)$ The relation is reflexive since $|a|=|a|$ is true

$(ii)$ The relation is symmetric as if $|a|=|b|$ then $|b|=|a|$

$(iii)$ The relation is transitive if $|a|=|b|$ and $|b|=|c|$ then it follows that $|a|=|c|$

And this is as far as I get. I am struggling when it comes to identifying an equivalence class for this relation. I honestly don't even think I'd know where to begin for this. A nudge in the right direction would be greatly appreciated.

Answer 1

If I've understood correctly, you need to characterize the equivalence classes for the above equivalence relation on the set $X=\mathbb{C}$.

The definition of complex absolute value is very geometric. Given some complex number $a+bi$, $\lvert a+bi\rvert=\sqrt{a^2+b^2}$. It follows from here that an equivalence class on $\mathbb{C}$ with the relation of absolute value is actually a circle of radius corresponding to the absolute value of elements in the class, centered at the origin.

So, multiple different classes form a collection of concentric circles about the origin. Can you see why?

Answer 2

What number(s) is $-3 + 4i$ equivalent to? What number(s) is $4.68$ equivalent to? What number(s) is $0$ equivalent to?

Answer 3

Hint:

You can find the answer if you think to complex number $a$ in polar form $a=|a| e^{i\theta}$. than you have an equivalence class for any value of $|a| \in \mathbb{R}$.