Cardinality/Examples/Powerset of Arbitrary Set

Example of Cardinality

Let:

$S_4 = \set {X \in \powerset {S_2}: \card X = 3}$

where $S_2 = \set {x \in \Z: 0 < x < 6}$.

The cardinality of $S_4$ is given by:

$\card {S_4} = 10$

Proof

By Cardinality of $S_2$, we have that:

$\card {S_2} = 5$

Then:

\(\ds \card {S_4}\)

\(=\)

\(\ds \dfrac {5!} {2! \, 3!}\)

Cardinality of Set of Subsets

\(\ds \)

\(=\)

\(\ds \dfrac {120} {2 \times 3}\)

Definition of Factorial

\(\ds \)

\(=\)

\(\ds \dbinom 5 3\)

Binomial Theorem

\(\ds \)

\(=\)

\(\ds 10\)

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 $4$