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In this link, Division of cardinals, someone asks a question about cardinal division and references a Wikipedia page about it. The Wikipedia page does not give a reference to their statement, but I'd really like to know one. Does anyone know specifically (preferably book and page number) where I can find this and a proof?

change_picture
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This follows from the fact that cardinal multiplication is not very interesting for infinite cardinals. Namely, if $\kappa$ and $\mu$ are infinite, $\kappa \cdot \mu = \max(\kappa,\mu)$. Thus, if we're given $\lambda$ and $\kappa$ we may always solve the equation in variable $\mu$ $$ \kappa \cdot \mu = \lambda $$ if and only if $\kappa \leq \lambda$. Indeed, if $\kappa \leq \lambda$, then $$\kappa \cdot \lambda = \max(\kappa,\lambda) = \lambda.$$ On the other hand, if $\kappa > \lambda$ $$\kappa \cdot \mu = \max(\kappa,\mu) \geq \kappa > \lambda.$$

juan diego rojas
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  • Thank you! This also has me thinking then, what can we say about the cardinality of a quotient set? That is, if we take a set of cardinality $\lambda$ and partition it into subsets each of cardinality $\kappa$, is it true then that the resulting quotient set has cardinality $\lamda/$ divided by $\kappa$, where both $\lamba$ and $\kappa$ are infinite cardinals with $\kappa \leq \lambda$? – change_picture Oct 15 '19 at 02:17
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    The cardinal $\lambda/\kappa$ is well defined (and equal to $\lambda$) only when $\kappa < \lambda$. When $\kappa = \lambda$, one has to be careful, as we may partition $\lambda$ in $\mu$ subsets (for every $\mu < \lambda$) each of cardinality $\lambda$. So, in that case, you need to calculate precisely the number of equivalence classes. – juan diego rojas Oct 15 '19 at 02:23
  • Yes of course. Thanks again! – change_picture Oct 15 '19 at 03:03