I was wondering about how to define the set of rational numbers, as I am currently learning about set theory in a class of mine. We were going through using set builders to define sets and produced two different ways of defining the set of Rational Numbers ($\mathbb{Q}$):
Definition 1: $\mathbb{Q} = \lbrace \frac{a}{b} \vert \ (a \in \mathbb{Z}) \ \wedge (b \in \mathbb{Z}) \ \wedge (b \not= 0) \rbrace$
Definition 2: $\mathbb{Q} = \lbrace x \vert \exists a,b \in \mathbb{Z}, \ x = \frac{a}{b} \rbrace$
Definition 1 introduced the concept of a "multiset," since multiple instances of the same element will occur (for example, $\frac{1}{2} = \frac{2}{4} = \frac{3}{6} = 0.5$) in the set defined by Definition 1.
My first question is whether Definition 2 would also produce a multiset.
It is my instinct that Definition 2 would not produce a multiset, since it is simply asking the question "Given some numerical value, are there two integers whose quotient equals that value?" and answering "If so, that value is in the set of rational numbers". Therefore, something like the value of $0.5$ (imagined purely as a value, a point on a number line for example) would only be included once since even though it can be described as the quotient of two integers in many different ways, the set is not asking for those different expressions but just the value that the expressions describe.
My second question is whether it is necessary to include the proposition $b \not= 0$ in Definition 2.
My instinct here is that this condition is not necessary for Definition 2, since it would be impossible to produce any value x by taking the quotient of an integer and $0$, so given any value $x$, there won't exist a quotient $\frac{a}{0}$ (where a is an integer) that will yield $x$.
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Thanks!