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View 2013_JUPITER-Z1__1N1DY460E1_.pdf from ACCOUNTING 2 at Batam International University. T115FSE (1DY7) T115FSEC (1DY8) T115FSEC.Q:
What is a set a given recursive set with # closure property?
I am somewhat confused as to how are sets (of formal language) defined that have the # closure property.
My first question is, for a given set $A$, what is $A$’s # closure property. Is it that, for any member $x \in A$ and for any $y \in A$, $x \# y \in A$?
Also, does the condition that A be recursive influence the existence of the # closure property?
A:
The # closure property is equivalent to the claim that, given $x,y \in A$, $x$ (and hence, by induction, $x$’s constant), should be in the set $A$, or that $x \# y \in A$.
So for a recursive set $A$, it is the case that for each $x \in A$ and for each $y \in A$, $x \# y \in A$, and hence, the # closure property holds.
This works for sets with # closure property for arbitrary $x$ and $y$. Since that requires that $A$ be recursive, the # closure property holds for sets with $A$ recursive.
If one wants to prove that $A$ has $y$’s # closure property for a particular $y$, it is possible to prove the claim by induction, but the induction proof is very different than the proof that was given here.
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