# vacuous

Suppose $X$ is a set and $P$ is a property defined as follows:

 $X$ has property $P$ if and only if $\forall Y[$ $Y$ satisfies condition $1]\Rightarrow$ $Y$ satisfies condition $2$

where condition $1$ and condition $2$ define the property. If condition $1$ is never satisfied then $X$ satisfies property $P$ vacuously.

## Examples

1. 1.

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].

2. 2.

The empty set is a Hausdorff space (vacuously).

3. 3.

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.

## References

• 1 Wikipedia http://en.wikipedia.org/wiki/Vacuous_truthentry on Vacuous truth.
Title vacuous Vacuous 2013-03-22 14:42:27 2013-03-22 14:42:27 matte (1858) matte (1858) 9 matte (1858) Definition msc 00A20 vacuously vacuously true vacuous truth