Cardinality/Examples/Powerset of Arbitrary Set
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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 $4$