# Definition:Basic Primitive Recursive Function/Projection Function

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## Definition

The **projection functions** $\pr_j^k: \N^k \to \N$ are basic primitive recursive functions, defined as:

- $\forall \tuple {n_1, n_2, \ldots, n_k} \in \N^k: \map {\pr_j^k} {n_1, n_2, \ldots, n_k} = n_j$
^{[1]}

where $j \in \closedint 1 k$.

They are each URM computable by a single-instruction URM program.

## Notation

- ↑ The usual notation for the projection function omits the superscript that defines the arity of the particular instance of the projection in question at the time, for example: $\pr_j$. However, in the context of computability theory, it is a
*very good idea*to be*completely certain*of*exactly*which projection function is under discussion.