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

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Take the indexed supremum:

$\ds \sup _{\map \Phi j} a_j$

where $\map \Phi j$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $\map \Phi j$ is true.

Then $\ds \sup_{\map \Phi j} a_j$ is defined as being $-\infty$.

This supremum is called a vacuous supremum.

This is because:

$\forall a \in \R: \sup \set {a, -\infty} = a$

Hence for all $j$ for which $\map \Phi j$ 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

Linguistic Note

The word vacuous literally means empty.

It derives from the Latin word vacuum, meaning empty space.