Cardinality/Examples
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Examples of Cardinality
Cardinality $3$
Let $S$ be a set.
Then $S$ has cardinality $3$ if and only if:
\(\ds \exists x: \exists y: \exists z:\) | \(\) | \(\ds x \in S \land y \in S \land z \in S\) | ||||||||||||
\(\ds \) | \(\land\) | \(\ds x \ne y \land x \ne z \land y \ne z\) | ||||||||||||
\(\ds \) | \(\land\) | \(\ds \forall w: \paren {w \in S \implies \paren {w = x \lor w = y \lor w = z} }\) |
That is:
- $S$ contains elements which can be labelled $x$, $y$ and $z$
- Each of these elements is distinct from the others
- Every element of $S$ is either $x$, $y$ or $z$.
Let:
\(\ds S_1\) | \(=\) | \(\ds \set {-1, 0, 1}\) | ||||||||||||
\(\ds S_2\) | \(=\) | \(\ds \set {x \in \Z: 0 < x < 6}\) | ||||||||||||
\(\ds S_3\) | \(=\) | \(\ds \set {x^2 - x: x \in S_1}\) | ||||||||||||
\(\ds S_4\) | \(=\) | \(\ds \set {X \in \powerset {S_2}: \card X = 3}\) | ||||||||||||
\(\ds S_5\) | \(=\) | \(\ds \powerset \O\) |
Cardinality of $S_1 = \set {-1, 0, 1}$
The cardinality of $S_1$ is given by:
- $\card {S_1} = 3$
Cardinality of $S_2 = \set {x \in \Z: 0 < x < 6}$
The cardinality of $S_2$ is given by:
- $\card {S_2} = 5$
Cardinality of $S_3 = \set {x^2 - x: x \in S_1}$
The cardinality of $S_3$ is given by:
- $\card {S_3} = 2$
Cardinality of $S_4 = \set {X \in \powerset {S_2}: \card X = 3}$
The cardinality of $S_4$ is given by:
- $\card {S_4} = 10$
Cardinality of $S_5 = \powerset \O$
The cardinality of $S_5$ is given by:
- $\card {S_5} = 1$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $4$