Definition:Summation/Propositional Function

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

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

Let $\map R j$ be a propositional function of $j$.

Then we can write the summation as:


 * $\ds \sum_{\map R j} a_j = \text{ The sum of all $a_j$ such that $\map R j$ 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 $\ds \sum_{j \mathop = 1}^n$ is merely another way of writing $\ds \sum_{1 \mathop \le j \mathop \le n}$.

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