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# 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. 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. The empty set is a Hausdorff space (vacuously).

3. Suppose property $P$ is defined by the statementâ€‰:

*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 entry on Vacuous truth.

## Mathematics Subject Classification

00A20*no label found*

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