Definition:Supremum of Set/Real Numbers/Propositional Function

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Definition

Let $\family {a_j}_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $\map R j$ be a propositional function of $j \in I$.


Then we can define the supremum of $\family {a_j}_{j \mathop \in I}$ as:

$\displaystyle \sup_{\map R j} a_j := \text{ the supremum of all $a_j$ such that $\map R j$ holds}$


If more than one propositional function is written under the supremum sign, they must all hold.


Finite Range

Let the fiber of truth of $\map R j$ be finite.

Then the supremum of $\family {a_j}_{j \mathop \in I}$ can be expressed as:

$\displaystyle \max_{\map R j} a_j = \text{ the maxmum of all $a_j$ such that $\map R j$ holds}$

and can be referred to as the maximum of $\family {a_j}_{j \mathop \in I}$.


Vacuous Supremum

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

  • Results about suprema can be found here.


Sources