Definition:Summation/Propositional Function

Definition
Let $\left({S, +}\right)$ be an algebraic structure where the operation $+$ is an operation derived from, or arising from, the addition operation on the natural numbers.

Let $\left({a_1, a_2, \ldots, a_n}\right) \in S^n$ be an ordered $n$-tuple in $S$.

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

Then we can write:


 * $\displaystyle \sum_{\Phi \left({j}\right)} a_j = \text{ The sum of all } a_j \text{ such that } \Phi \left({j}\right) \text{ holds}$.

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

Such an operation on an ordered tuple is known as a summation.

Note that the definition by inequality form $1 \le j \le n$ is a special case of such a propositional function.

Also note that the definition by index form $\displaystyle \sum_{j \mathop = 1}^n$ is merely another way of writing $\displaystyle \sum_{1 \mathop \le j \mathop \le n}$.

Hence all instances of a summation can be expressed in terms of a propositional function.