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:

\(\displaystyle \exists x: \exists y: \exists z:\) \(\) \(\displaystyle x \in S \land y \in S \land z \in S\)
\(\displaystyle \) \(\land\) \(\displaystyle x \ne y \land x \ne z \land y \ne z\)
\(\displaystyle \) \(\land\) \(\displaystyle \forall w: \paren {w \in S \implies \paren {w = x \lor w = y \lor w = z} }\)


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:

\(\displaystyle S_1\) \(=\) \(\displaystyle \set {-1, 0, 1}\)
\(\displaystyle S_2\) \(=\) \(\displaystyle \set {x \in \Z: 0 < x < 6}\)
\(\displaystyle S_3\) \(=\) \(\displaystyle \set {x^2 - x: x \in S_1}\)
\(\displaystyle S_4\) \(=\) \(\displaystyle \set {X \in \powerset {S_2}: \card X = 3}\)
\(\displaystyle S_5\) \(=\) \(\displaystyle \powerset \O\)


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