Show that $[a][\star][b] :=[a \star b]$ defines an operation on $G/\sim$.

Question

Let $G$ be a set equipped with an operation $\star$ and an equivalence relation $\sim$. Suppose that $\sim$ is compatible with $\star$, i.e., for elements $a$, $a'$, $b$, $b'$ of $G$, $$\text{if}\ a \sim a' \ \text{and}\ b\sim b', \ \text{then}\ a\star b \sim a' \star b'.$$ For $a \in G$, denote the $\sim$-equivalence class of $a$ by $[a]$, and denote the quotient set of $G$ with respect to $\sim$ by $G/\sim$.

Show that $$[a][\star][b] :=[a \star b]$$ defines an operation on $G/\sim$.

I know that an operation would be $G/\sim \times \space G/\sim$ $\rightarrow$ $G/\sim$. Do I just have to show closure? How would I do that with the operation $[\star]$

Answer 1

Hint. You have to show that the operation is well-defined, that is

if $[a]=[a']$ and $[b]=[b']$ then $[a\star b]=[a'\star b']$.

This is pretty much what you have been given - really, all you have to do is rewrite it in a different notation.

You also have to note that (obviously?) $[a\star b]$ is an element of $G\,/\,\mathord{\sim}$.

Good luck!