Does the set of rational numbers include repeating values?

Question

In my Discrete Math lecture, we were told that we should omit any number that repeats in the set of rational numbers $\mathbb{Q}$. In other words, we should only keep $\frac{1}{1}$ and omit $\frac{2}{2}, \frac{3}{3}$, etc. But I don't see why this would be the case, since the set is defined as $\mathbb{Q} = \{\frac{a}{b}|a,b\in\mathbb{N}, b\ne 0\} $ Does the set $\mathbb{Q}$ include these repeating values?

Answer 1

That is a nice question. You see, when we build the set $\mathbb{Q}$ we do not only define the elements in form, there is an additional definition of "equality". Let $x=\frac{a}{b}$ and $y=\frac{c}{d}$. We say that $$x=y\Leftrightarrow ad=bc$$

So in fact, the set of rational numbers is the set $$\left\{\frac{a}{b}\vert a\in \mathbb{Z}\wedge b\in \mathbb{Z}^\ast\right\}$$ after we identify i.e. "glue" the equal numbers together.