# Definition:Primitive Recursive

## Definition

### Function

A function is primitive recursive if it can be obtained from basic primitive recursive functions using the operations of substitution and primitive recursion a finite number of times.

### Set

Let $A \subseteq \N$.

Then $A$ is a primitive recursive set iff its characteristic function $\chi_A$ is a primitive recursive function.

### Relation

Let $\mathcal R \subseteq \N^k$ be an $n$-ary relation on $\N^k$.

Then $\mathcal R$ is a primitive recursive relation if and only if its characteristic function $\chi_\mathcal R$ is a primitive recursive function.