(a) Show that $\sim$ is an equivalence relation on $\mathbb{N}$.
(b) Describe the equivalence classes [3], [9], and [99].
(c) If $a\sim b$, which attributes of $a \text{ and } b$ are equal?
For (a) I have to show that $\sim$ is reflexive, transitive, and symmetric in order for it to be an equivalence relation.
So if $ab$ is square, $a = b$ so the relation is reflexive. Next, take $a\sim b$ and $b \sim c$, because $ab$ is square and $bc$ is square, $a = b = c$, so $a = c$. Thus $\sim$ is transitive. For symmetric, I'm not sure. But before I continue, can I even say that if $ab$ is square, $a = b$. I was thinking of $16 = 2 \cdot 8$ which seems like $a$ and $b$ are not equal but is that just a square number or a square? Is there a difference?
If I can't do that, how am I supposed to go about it?
Best Answer
I’ll get you started. To show that $\sim$ is reflexive, you must show that if $n\in\Bbb N$, then $n\sim n$. Check the definition of $\sim$: this means that $n\cdot n$ is a square. Of course $n\cdot n=n^2$ is a square, so $n\sim n$, and $\sim$ is reflexive. You should have no trouble showing that $\sim$ is symmetric. For transitivity, suppose that $k\sim m$ and $m\sim n$. Then $km$ and $mn$ are squares, say $km=a^2$ and $mn=b^2$; you must show that $k\sim n$, i.e., that $kn$ is a square. Try to write $kn$ in terms of the pieces that you already have, doing it in a way that demonstrates that $kn$ is a square.
$[3]$ is by definition the set of all $n\in\Bbb N$ such that $3\sim n$, i.e., such that $3n$ is a square. What does this tell you about $n$? HINT: What can you say about the number of factors of $3$ in the prime factorization of $n$? Thinking in similar terms will get you through the rest of (b) as well.
For (c) you should be thinking about the prime factorizations of $a$ and $b$.