# Set Union/Examples

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## Contents

- 1 Examples of Set Union
- 1.1 Example: $2$ Arbitrarily Chosen Sets
- 1.2 Example: $2$ Arbitrarily Chosen Sets of Complex Numbers
- 1.3 Example: $3$ Arbitrarily Chosen Sets
- 1.4 Example: $3$ Arbitrarily Chosen Sets of Complex Numbers
- 1.5 Example: People who are Blue-Eyed or British
- 1.6 Example: Overlapping Integer Sets
- 1.7 Example: Subset of Union
- 1.8 Example: $2$ Circles in Complex Plane
- 1.9 Example: $3$ Circles in Complex Plane

## Examples of Set Union

### Example: $2$ Arbitrarily Chosen Sets

Let:

\(\displaystyle S\) | \(=\) | \(\displaystyle \set {a, b, c}\) | |||||||||||

\(\displaystyle T\) | \(=\) | \(\displaystyle \set {c, e, f, b}\) |

Then:

- $S \cup T = \set {a, b, c, e, f}$

### Example: $2$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {3, -i, 4, 2 + i, 5}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {-i, 0, -1, 2 + i}\) |

Then:

- $A \cup B = \set {3, -i, 0, -1, 4, 2 + i, 5}$

### Example: $3$ Arbitrarily Chosen Sets

Let:

\(\displaystyle A_1\) | \(=\) | \(\displaystyle \set {1, 2, 3, 4}\) | |||||||||||

\(\displaystyle A_2\) | \(=\) | \(\displaystyle \set {1, 2, 5}\) | |||||||||||

\(\displaystyle A_3\) | \(=\) | \(\displaystyle \set {2, 4, 6, 8, 12}\) |

Then:

- $A_1 \cup A_2 \cup A_3 = \set {1, 2, 3, 4, 5, 6, 8, 12}$

### Example: $3$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {1, i, -i}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {2, 1, -i}\) | |||||||||||

\(\displaystyle C\) | \(=\) | \(\displaystyle \set {i, -1, 1 + i}\) |

Then:

- $\paren {A \cup B} \cup C = \set {2, 1, -i,1, 1 + i}$

### Example: People who are Blue-Eyed or British

Let:

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {\text {British people} }\) | |||||||||||

\(\displaystyle C\) | \(=\) | \(\displaystyle \set {\text {Blue-eyed people} }\) |

Then:

- $B \cup C = \set {\text {People who are blue-eyed or British or both} }$

### Example: Overlapping Integer Sets

Let:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {x \in \Z: 2 \le x}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {x \in \Z: x \le 5}\) |

Then:

- $A \cup B = \Z$

### Example: Subset of Union

Let $U, V, W$ be non-empty sets.

Let $W$ be such that for all $w \in W$, either:

- $w \in U$

or:

- $w \in V$

Then:

- $W \subseteq U \cup V$

### Example: $2$ Circles in Complex Plane

Let $A$ and $B$ be sets defined by circles embedded in the complex plane as follows:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {z \in \C: \cmod {z - 1} < 3}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {z \in \C: \cmod {z - 2 i} < 2}\) |

Then $A \cup B$ can be illustrated graphically as:

where the union is depicted in yellow.

### Example: $3$ Circles in Complex Plane

Let $A$, $B$ and $C$ be sets defined by circles embedded in the complex plane as follows:

\(\displaystyle A\) | \(=\) | \(\displaystyle \set {z \in \C: \cmod {z + i} < 3}\) | |||||||||||

\(\displaystyle B\) | \(=\) | \(\displaystyle \set {z \in \C: \cmod z < 5}\) | |||||||||||

\(\displaystyle C\) | \(=\) | \(\displaystyle \set {z \in \C: \cmod {z + 1} < 4}\) |

Then $A \cup B \cup C$ can be illustrated graphically as:

where the union is depicted in yellow.