ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Collection of sets
Set union
Successor set
Inductive set
Set of inductive sets
Set of natural numbers
Set of integers
Definition D368
Set of rational numbers
Formulation 0
Let $\mathbb{Z}$ be the D367: Set of integers such that
(i) $\cdot : \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z}$ is the D608: Integer multiplication operation
(ii) \begin{equation} \mathbb{Z}_{\neq 0} : = \mathbb{Z} \setminus \{ 0 \} \end{equation}
(iii) \begin{equation} {\sim} : = \left\{ ((a, b), (c, d)) \in (\mathbb{Z} \times \mathbb{Z}_{\neq 0})^2 : a \cdot d = b \cdot c \right\} \end{equation}
The set of rational numbers is the D180: Quotient set \begin{equation} \mathbb{Q} : = (\mathbb{Z} \times \mathbb{Z}_{\neq 0}) / {\sim} \end{equation}
Rational number
Set of euclidean rational numbers