Introduction [[Set Theory]] [[Proof by Induction]] [[Function]] [[Image and Inverse Image]] [[Injective Function]] [[Surjective Function]] [[Bijective Function]] [[Composition of Functions]] [[Binary Relation]] [[Reflexive Property]] [[Symmetric Property]] [[Transitive Property]] [[Equivalence Relation]] [[Equivalence Class]] [[Every element under an equivalence relation is in exactly one equivalence class]] The set of [[rational number|rational numbers]] $\frac{a}{b}$ can be defined as [[Equivalence Class|equivalence classes]] $[[a,b)](a,b|(a,b)]]) $a$ and a [[Natural Number|natural number]] $b$, that is elements of $\mathbb{Z} \times \mathbb{N}$. The relation is defined by $(a, b) \sim(c, d)$ whenever $a d=b c$. It is an [[Equivalence Relation|equivalence relation]] because: $\begin{array}{ccll} (a, b) \sim(a, b)& &&\text{Reflexive}\\ (a, b) \sim(c, d)& \iff& (c, d) \sim(a, b) & \text{Symmetric}\\ (a, b) \sim(c, d) \land (c, d) \sim(m, n) &\implies& (a,b)\sim(m,n) & \text{Transitive} \end{array}$ [[Cardinality]]