# Prove $B = \{ \varphi_n(n) | n \in \mathsf{K} \}$ to be recursive

The set $B$ is the range of universal function given the domain $\mathsf{K}$, where $\mathsf{K} = \{ n | \varphi_n(n) \textit{ halts} \}$. How can we prove such claim?

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Don't use the title and the question box has one. Write the whole question in the question box. – Git Gud Jan 12 '13 at 18:11
Presumably you want to know whether $B$ is recursive relative to $K$, i.e. $B \leq_\mathrm{T} K$, not whether it is plain recursive. – Benedict Eastaugh Jan 12 '13 at 18:14
I would like to know how to prove $B$ is "plain" recursive. – MeadowMuffins Jan 12 '13 at 18:19

You might find it easier to prove the stronger result that $B$ contains all of the natural numbers. (A sneaky question)
How can you be sure that $B \leftrightarrow \mathbb{N}$? – MeadowMuffins Jan 12 '13 at 20:42
@MeadowMuffins Let $k \in \omega$. Let $\Psi_k$ denote the constant function taking value $k$. $\Psi_k$ is clearly computable. Hence $\Psi_k = \varphi_n$ for some $n$. Since $\Psi_k$ is the constant function $\varphi_n(n) \downarrow$ so $n \in K$. Moreover, $\varphi_n(n) = \Psi_k(n) = k$. So $B = \omega$. – William Jan 12 '13 at 23:24