Definition:Supremum of Set/Real Numbers/Propositional Function

Definition
Let $\left \langle {a_j} \right \rangle_{j \mathop \in I}$ be a family of elements of the real numbers $\R$ indexed by $I$.

Let $R \left({j}\right)$ be a propositional function of $j \in I$.

Then we can write the supremum of $\left \langle {a_j} \right \rangle_{j \mathop \in I}$ as:


 * $\displaystyle \sup_{R \left({j}\right)} a_j = \text{ The supremum of all $a_j$ such that $R \left({j}\right)$ holds}$.

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