# Cardinality/Examples

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## Contents

## 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

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $4$