How to prove equivalence relations

Question

I'm going through Pinter's "A Book of Abstract Algebra" and I'm currently on the topic of Partitions and Equivalence Relations. I'm having a little trouble understanding the way he (and apparently many other authors) word their questions. For instance, he asks:

Prove that each of the following is an equivalence relation on the indicated set. Then describe the partition associated with the equivalence relation.

On $ \mathbb{Z}$, show that $a\cong b$ iff $|a|$ = $|b|$.

Which is simple enough, but I'm confused by the 'iff' part. I know how I'm supposed to prove something like this, but I'm often not confident because I thought that I'm supposed to be proving two conditional statements:

If $a\cong b$, then $|a|$ = $|b|$, and If $|a|$ = $|b|$, then $a\cong b$.

I know how to prove the second conditional and apparently that's the only one I'm supposed to prove, but then what's the point of the 'iff'? Is there something I'm not understanding?

Thanks in advance!

Answer 1

In my copy of Pinter, he actually asks:

Prove that each of the following is an equivalence relation on the indicated set. Then describe the partition associated with that equivalence relation.

1. On $\mathbb{Z}$, $m\cong n$ iff $|m| = |n|$

Pinter is defining what $m\cong n$ means and asking you to show that $\cong$ is an equivalence relation.

The generic proof for an equivalence relation looks like:

• Reflexivity: Let $a\in \text{[whatever set]}$. Then [chain of logic based on definition of $\cong$], therefore $a \cong a$.
• Symmetry: Assume $a \cong b$, i.e. [spell out what this means]. Then [chain of logic], i.e. $b \cong a$.
• Transitivity: Assume $a \cong b$ and $b \cong c$, i.e. [spell out what this means]. Then [chain of logic], i.e. $a \cong c$.

E.g. the proof for your example would look like:

• Reflexivity: Let $a\in\mathbb{Z}$. Then [stuff you write], therefore $a \cong a$.
• Symmetry: Assume $a \cong b$, i.e. $|a| = |b|$. Then [stuff you write], i.e. $b \cong a$.
• Transitivity: Assume $a \cong b$ and $b \cong c$, i.e. $|a| = |b|$ and $|b| = |c|$. Then [stuff you write], i.e. $a \cong c$.

As a very strong hint, the last step before you write "therefore $x \cong y$" will be "$|x| = |y|$, therefore $x \cong y$".

Try doing it yourself first. Spoiler: This question contains an answer, though I think it's awkwardly worded.