Cardinality/Examples/Powerset of Empty Set

Example of Cardinality

Let:

$S_5 = \powerset \O$

where:

$\O$ denotes the empty set

$\powerset \O$ denotes the power set of $\O$.

The cardinality of $S_5$ is given by:

$\card {S_5} = 1$

Proof

By Power Set of Empty Set, we have that:

$\powerset \O = \set {\O}$

Thus $\powerset \O$ contains exactly $1$ element, that is: $\O$.

Hence the result by definition of cardinality.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $4$