Definition:Vacuous

Disambiguation

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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 product notation over an empty set of multiplicands. Such a product has a value of one (not zero).

Idle; having no import.

Linguistic Note

The word vacuous literally means empty.

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

Also see

• Definition:Empty
• Definition:Trivial

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)