How would I specify a set of elements that belong to an equivalent class, given that the elements need to be unique in a set?

Question

Consider for example that a rational number can be specified as $q = \frac{a}{b}$ where $a,b \in \mathbb{Z}, b \ne 0$.

If we now consider two rationals to be equivalent if they have the same sign and magnitude, then the rationals $\frac{2}{3}$ and, say, $\frac{6}{9}$ are the same.

If I want to specify the rationals, I might state the set as:

$$\mathbb{Q} = \{\frac{a}{b} | a, b \in \mathbb{Z}, b\ne 0\}$$

which is the rule I used above. Implicit in this definition, though, is that if two rationals are equal in the sense of sign and magnitude, then only one is chosen and the set elements are unique.

1. How do I state that explicitly in my set builder notation?
2. What if I instead wanted all possible fractions that represent the same rational number? How do I write that and ensure that the uniqueness of the elements in my set is communicated?

Answer 1

Your set is not just a set of numbers but a set of "symbols" that represent numbers. If we were talking about a set of numbers then $\{1/2,1/4, 1/8,\cdot\cdot\cdot\}$ or $\{\frac{a}{b}\}$ are bad notation but if you are talking about a set of symbols then those are valid notations.