# Definition:Continued Product/Vacuous Product

< Definition:Continued Product(Redirected from Definition:Vacuous Product)

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## Definition

Take the composite expressed as a continued product:

- $\ds \prod_{\map R j} a_j$

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

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

Then $\ds \prod_{\map R j} a_j$ is **defined** to be $1$.

**Beware:** *not* zero.

This composite is called a **vacuous product**.

This is because:

- $\forall a: a \times 1 = a$

where $a$ is a number.

Hence for all $j$ for which $\map R j$ is false, the value of the product is unaffected.

This is most frequently seen in the form:

- $\ds \prod_{j \mathop = m}^n a_j = 1$

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 $n$.

## 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 $23$