Definition:Supremum of Set/Real Numbers/Propositional Function/Vacuous Supremum

Definition
Take the indexed supremum:
 * $\displaystyle \sup _{\Phi \left({j}\right)} a_j$

where $\Phi \left({j}\right)$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $\Phi \left({j}\right)$ is true.

Then $\displaystyle \sup_{\Phi \left({j}\right)} a_j$ is defined as being $-\infty$.

This supremum is called a vacuous supremum.

This is because:
 * $\forall a \in \R: \sup \left\{ {a, -\infty}\right\} = a$

Hence for all $j$ for which $\Phi \left({j}\right)$ is false, the supremum is unaffected.

In this context $-\infty$ is considered as minus infinity, the hypothetical quantity that has the property:


 * $\forall n \in \Z: -\infty< n$

Also see

 * Definition:Vacuous Truth
 * Definition:Vacuous Summation
 * Definition:Vacuous Product