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

## 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

- 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 $35$