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Suppose $X$ is a set and $P$ is a property defined as follows:
where condition $1$ and condition $2$ define the property. If condition $1$ is never satisfied then $X$ satisfies property $P$ vacuously.
- If $X$ is the set $\{1,2,3\}$ and $P$ is the property defined as above with condition $1=$ $Y$ is a infinite subset of $X$ , and condition $2=$ $Y$ contains $7$ . Then $X$ has property $P$ vacously; every infinite subset of $\{1,2,3\}$ contains the number $7$ [1].
- The empty set is a Hausdorff space (vacuously).
- Suppose property $P$ 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.
- 1
- Wikipedia entry on Vacuous truth.
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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, number, contains, infinite subset, property
There are 9 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 4775 times total.
Classification:
| AMS MSC: | 00A20 (General :: General and miscellaneous specific topics :: Dictionaries and other general reference works) |
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Pending Errata and Addenda
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![$\displaystyle \mbox{$\forall Y[$\ $Y$\ satisfies condition $1] \Rightarrow$\ $Y$\ satisfies condition $2$\ }$](http://images.planetmath.org:8080/cache/objects/6326/js/img2.png "$\displaystyle \mbox{$\forall Y[$\ $Y$\ satisfies condition $1] \Rightarrow$\ $Y$\ satisfies condition $2$\ }$"){kind=small}