# Category:Definitions/Inductive Sets

This category contains definitions related to Inductive Sets.
Related results can be found in Category:Inductive Sets.

Let $S$ be a set of sets.

Then $S$ is inductive if and only if:

 $(1)$ $:$ $S$ contains the empty set: $\ds \quad \O \in S$ $(2)$ $:$ $S$ is closed under the successor mapping: $\ds \forall x:$ $\ds \paren {x \in S \implies x^+ \in S}$ where $x^+$ is the successor of $x$ That is, where $x^+ = x \cup \set x$

## Pages in category "Definitions/Inductive Sets"

The following 4 pages are in this category, out of 4 total.