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A function is primitive recursive if and only if it can be obtained from basic primitive recursive functions using the operations of substitution and primitive recursion a finite number of times.
Let $A \subseteq \N$.
Then $A$ is a primitive recursive set if and only if its characteristic function $\chi_A$ is a primitive recursive function.
Let $\RR \subseteq \N^k$ be an $n$-ary relation on $\N^k$.
Then $\RR$ is a primitive recursive relation if and only if its characteristic function $\chi_\RR$ is a primitive recursive function.