Definition:Summation/Vacuous Summation

Definition

Take the summation:

$\ds \sum_{\map \Phi j} a_j$

where $\map \Phi j$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $\map \Phi j$ is true.

Then $\ds \sum_{\map \Phi j} a_j$ is defined as being $0$.

This summation is called a vacuous summation.

This is because:

$\forall a: a + 0 = a$

where $a$ is a number.

Hence for all $j$ for which $\map \Phi j$ is false, the sum is unaffected.

This is most frequently seen in the form:

$\ds \sum_{j \mathop = m}^n a_j = 0$

where $m > n$.

In this case, $j$ can not at the same time be both greater than or equal to $m$ and less than or equal to $n$.

Some sources consider such a treatment as abuse of notation.

A vacuous summation can also be referred to as a vacuous sum or empty sum.

The word vacuous literally means empty.

It derives from the Latin word vacuum, meaning empty space.