PlanetMath: vacuous

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vacuous

(Definition)

Suppose is a set and is a property defined as follows:

		$% latex2html id marker 277 $\displaystyle \mbox{$X$\ has property $P$\ if and only if}$$
		$\displaystyle \mbox{$\forall Y[$\ $Y$\ satisfies condition $1] \Rightarrow$\ $Y$\ satisfies condition $2$\ }$

where condition

and condition

define the property. If condition

is never satisfied then

satisfies property

vacuously.

If is the set $\{1,2,3\}$ and is the property defined as above with condition is a infinite subset of , and condition contains . Then has property vacously; every infinite subset of $\{1,2,3\}$ contains the number [1].
The empty set is a Hausdorff space (vacuously).
Suppose property is defined by the statement:
The present King of France does not exist.
Then either of the following propositions is satisfied vacuously.
The present king of France is bald.
The present King of France is not bald.

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"vacuous" is owned by matte. [ full author list (4) ]

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Other names:

vacuously, vacuously true, vacuous truth

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Cross-references: propositions, Hausdorff space, empty set, contains, infinite subset, property
There are 8 references to this entry.

This is version 6 of vacuous, born on 2004-10-09, modified 2008-02-23.
Object id is 6326, canonical name is Vacuous2.
Accessed 2954 times total.

Classification:

00A20 (General :: General and miscellaneous specific topics :: Dictionaries and other general reference works)

Pending Errata and Addenda

None.

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