Is ~ symmeteric?

Question

I was given the following problem in my assignment:

Define $a$~$b$ on the rationals by $a$~$b$ iff $b=ak^2$ for some rational number $k$.

Is ~ symmetric?.

Please somebody explain to me how to do this problem. Thanks!

Answer 1

To show that a relation $R$ is symmetric, you need to show that if $aRb$ then $bRa$. So suppose $a\sim b$. That means that there is some rational $k$ such that $b=ak^2$. You want to show that $b\sim a$, so: $$\mbox{Can you find a rational $l$ such that $a=bl^2$?}$$

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To show that a relation $R$ is not symmetric, you need to find a counterexample: some $a, b$ such that $aRb$ but $b\not Ra$. Can you do this in this case? That is, can you find some $a, b$ such that

• there is a rational $k$ with $b=ak^2$, but

• there is no rational $l$ with $a=bl^2$?

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Note that I haven't told you what the answer is - I've just described the strategy for both of the possible answers. You need to think about which strategy makes sense for this particular problem.

HINT: given $b=ak^2$, can you solve for $a$?