Description:
[218 B] implies Form 218
Content:
In Rubin, H./Rubin, J. [1985] pages
122--124, theorems 6.35 through 6.37 it is shown that forms [218 A] and [218 B]
are equivalent. It is clear that Form 218 (\(\forall n \in \Bbb N- \{ 0,1\},MC(\infty,\infty,\) relatively prime to \(n\))) implies
[218 B] (For all prime numbers \(p, MC(\infty,\infty,\) relatively prime to \(p\))) . In this note we argue that [218 B] implies Form 218.
Assume that \(n \in \Bbb N - \{0,1 \}\) and that \(X\) is a family of
non-empty sets. Let
\[X' = X \cup \{ y : y \hbox{ is finite and non-empty and } \exists z \in X \hbox{ such that } y \subseteq z \}.\]
Let \(\{ p_i: 1 \le i \le k \}\) be the set of prime divisors of \(n\). By [218 B],
there are functions \(f_i, 1 \le i \le k\) such that for \(1 \le i \le k\), \(f_i\) is a function with domain \(X'\) and for all \(y \in X'\),
\(f_i(y)\) is a finite non-empty subset of \(y\) such that \(|f_i(y)|\) and \(p_i\) are relatively prime. Let \(f = f_1 \circ f_2 \circ \cdots \circ f_k\)
then for \(y\in X\), one of the sets \(f(y), f(f(y)) = f^2(y), f^3(y), \ldots\) will have cardinality relatively prime to \(p_i\) for all \(i\), \(1\le i \le k\) and
therefore will have cardinality relatively prime to \(n\). Define \(g(y) = f^r(y)\) where \(r\) is the least natural number for which \(|f^r(y)|\) is relatively prime to \(n\). The function \(g\) is the function required by
the conclusion of Form 218 .
Howard-Rubin number:
161
Type:
Equivalences
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