# Definition:Inductive Class/General

## Definition

Let $A$ be a class.

Let $g: A \to A$ be a mapping on $A$.

Then $A$ is inductive under $g$ if and only if:

 $(1)$ $:$ $A$ contains the empty set: $\ds \quad \O \in A$ $(2)$ $:$ $A$ is closed under $g$: $\ds \forall x:$ $\ds \paren {x \in A \implies \map g x \in A}$

## Also see

• Results about inductive classes can be found here.