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 $x^+ = x \cup \set x$