# Category:Definitions/Inductive Classes

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This category contains definitions related to Inductive Classes.

Related results can be found in **Category:Inductive Classes**.

Let $A$ be a class.

Then $A$ is **inductive** if and only if:

\((1)\) | $:$ | $A$ contains the empty set: | \(\ds \quad \O \in A \) | ||||||

\((2)\) | $:$ | $A$ is closed under the successor mapping: | \(\ds \forall x:\) | \(\ds \paren {x \in A \implies x^+ \in A} \) | where $x^+$ is the successor of $x$ | ||||

That is, where $x^+ = x \cup \set x$ |

## Subcategories

This category has the following 4 subcategories, out of 4 total.

## Pages in category "Definitions/Inductive Classes"

The following 8 pages are in this category, out of 8 total.