I think I understand what a binary operation is, but I want to check my understanding with this example to be sure.
First of all, my textbook defines a binary operation in the following way:
Let $A$ be a set. A function from $A \times A$ to $A$ is called a binary operation on $A$.
Now for my question:
Define $*$ like this: For real numbers $a$, $b$, and $c$, $c=a*b$ if $a^2c^2=b^2$. Is $*$ a binary operation? Why or why not?
My initial sense is that $*$ is not a binary operation because we can re-arrange the equation as $c = \pm \sqrt{\frac{b^2}{a^2}}$. So $c$ is not uniquely determined by $a$ and $b$. Since a function from $\mathbb{R} \times \mathbb{R}$ to $\mathbb{R}$ must have a unique output, this means that $*$ cannot be properly described by a function and is thus not a binary operation.
Best Answer
Your argument is correct. Your expression $a^2c^2=b^2$ does not express $c$ in a unique way as a function of $a$ and $b$. Therefore, you don't have a function from $\mathbb R\times\mathbb R$ into $\mathbb R$, which means that you don't have a binary operation.