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$\overline{K}\;\leq\;\{ \langle p,q\rangle \mid |\mathtt{Dom}(\varphi_p)|=|\mathtt{Dom}(\varphi_q)|=|\mathtt{Im}(\varphi_p)|=|\mathtt{Im}(\varphi_q)|=1\;\wedge\;|\mathtt{Dom}(\varphi_p)\cup\mathtt{Dom}(\varphi_q)\cup\mathtt{Im}(\varphi_p)\cup\mathtt{Im}(\varphi_q)|=1 \}$
Reduce $\overline{K}$ to the set of pairs of natural numbers codifying programs
such that each domain and image of the function computed by each of such
programs has exactly $1$ element, and the union of the domains and images of
both programs has exactly $1$ element (roughly, the set of pairs of programs
whose domains and images have exactly $1$ element, and the union of all such
domains and images have exactly $1$ element) in order to prove that such set is
not semi-decidable (not recursively enumerable).
Authors: Carles Creus, Guillem Godoy
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