Definition:Supremum of Set/Real Numbers/Propositional Function/Finite Range
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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 $j \in I$.
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:
- $\ds \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.
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 $35$