Check if given transformations are operations

Question

I am to check whether given transformations on given sets are operations.
There is an example:
$\mathbb{Q}, a\circ b = a + b - ab$.
I would appreciate a solution. I'm new to this kind of Algebra and I do not know how to solve this problem.

Answer 1

Yes, it's an operation because for all $\{a,b\}\subset\mathbb Q$ we have $a+b-ab\in\mathbb Q$.

Indeed, $$\mathbb Q=\left\{\frac{m}{n}|m\in\mathbb Z,n\in\mathbb N\right\}.$$

Thus, there are $\{m_1,m_2\}\subset\mathbb Z$ and $\{n_1,n_2\}\subset\mathbb N$, for which $$a=\frac{m_1}{n_1}$$ and $$b=\frac{m_2}{n_2}.$$ Thus, $$a+b-ab=\frac{m_1}{n_1}+\frac{m_2}{n_2}-\frac{m_1m_2}{n_1n_2}=\frac{m_1n_2+m_2n_1-m_1m_2}{n_1n_2}\in\mathbb Q.$$