Is the following ~ transitive?

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 ~ transitive?.

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

Answer 1

To prove transitivity, we must show that $\forall a,b,c\in \mathbb{Q}$, the relations $a \sim b$ and $b \sim c$ imply $a \sim c$.

If $a \sim b$, we know there exists some $k_1 \in \mathbb{Q}$ such that $b = ak_1^2$ by definition. Additionally, if $b \sim c$ then there exists some $k_2 \in \mathbb{Q}$ such that $c = bk_2^2$. Therefore, via substitution, we find that these two conditions imply $c = ak_1^2k_2^2 = a(k_1k_2)^2$; Since $k_1k_2 \in \mathbb{Q}$ by closure under multiplication, we find that $a\sim c$ is implied; in this case the $k$ satisfying $c=ak^2$ would be $k_1 \cdot k_2$