Definition:Supremum of Set/Real Numbers/Propositional Function/Finite Range

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 $j \in I$.

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 $\family {a_j}_{j \mathop \in I}$.

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