Definition:Supremum of Set/Real Numbers/Propositional Function/Vacuous Supremum
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Definition
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.
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $35$