# Definition:Summation/Vacuous Summation

## Definition

Take the summation:

- $\displaystyle \sum _{\Phi \left({j}\right)} a_j$

where $\Phi \left({j}\right)$ is a propositional function of $j$.

Suppose that there are no values of $j$ for which $\Phi \left({j}\right)$ is true.

Then $\displaystyle \sum_{\Phi \left({j}\right)} a_j$ 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 $\Phi \left({j}\right)$ is false, the sum is unaffected.

This is most frequently seen in the form:

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

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.

## Also known as

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

## Also see

## Linguistic Note

The word **vacuous** literally means **empty**.

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

## Sources

- 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products - 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 $23$