# Definition:Vacuous

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## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Vacuous** may refer to:

- Logic:
- A vacuous truth is a statement about a collection of objects none of which exist. By definition, such a statement is understood to be true.

- Set Theory:
- The vacuous set is the same thing as the empty set, that is, the set with no elements. This Empty Set is Unique.

- Algebra:
- A vacuous summation is a summation over an empty set of summands. Such a sum has a value of
**zero**. - A vacuous product is an expression in the context of a continued product over an empty set of multiplicands. Such a continued product has a value of
**one**(not zero).

- A vacuous summation is a summation over an empty set of summands. Such a sum has a value of

*Idle; having no import.*

## Linguistic Note

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

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

## Also see

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**empty**:**2.** - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**vacuous**(of an operator or an expression)