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In my discrete mathematics class, we came across the definition of well-ordered for sets:

A set is well-ordered with respect to an ordering function if, $\forall $of its nonempty subsets, $\exists$ a minimum element.

My thought was that the empty set, $\emptyset$, would fulfill this definition, albeit "vacuously" since it has no nonempty subsets. However, I was told that this is incorrect and am not sure why. Could someone please elucidate whether or not this is true and how I can determine if a definition is fulfilled "vacuously?"

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  • $\begingroup$ $0$ is an ordinal number after all. $\endgroup$ – Anon Sep 19 '16 at 16:23
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The only binary relation on $\varnothing$ is $\varnothing$, since that’s the only subset of $\varnothing\times\varnothing$. It’s vacuously reflexive, antisymmetric, transitive, and total, so it’s a linear order on $\varnothing$. And as you say, it’s vacuously true that every non-empty subset of $\varnothing$ has a least element in this order, since $\varnothing$ has no non-empty subset, so it is in fact a well-order on $\varnothing$. Whoever told you otherwise was simply wrong.

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  • $\begingroup$ Thank you for your answer! $\endgroup$ – user322548 Sep 19 '16 at 16:04
  • $\begingroup$ @Ethan: You’re welcome! $\endgroup$ – Brian M. Scott Sep 19 '16 at 16:04

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