ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
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}
Children
Rational number
Set of euclidean rational numbers