Cardinality/Examples

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 $4$