# What does it mean to fulfill a definition vacuously?

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?"

• $0$ is an ordinal number after all. – Anon Sep 19 '16 at 16:23

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.