Set Union/Examples
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Examples of Set Union
Example: $2$ Arbitrarily Chosen Sets
Let:
| \(\ds S\) | \(=\) | \(\ds \set {a, b, c}\) | ||||||||||||
| \(\ds T\) | \(=\) | \(\ds \set {c, e, f, b}\) |
Then:
- $S \cup T = \set {a, b, c, e, f}$
Example: $2$ Arbitrarily Chosen Sets of Complex Numbers
Let:
| \(\ds A\) | \(=\) | \(\ds \set {3, -i, 4, 2 + i, 5}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {-i, 0, -1, 2 + i}\) |
Then:
- $A \cup B = \set {3, -i, 0, -1, 4, 2 + i, 5}$
Example: $3$ Arbitrarily Chosen Sets
Let:
| \(\ds A_1\) | \(=\) | \(\ds \set {1, 2, 3, 4}\) | ||||||||||||
| \(\ds A_2\) | \(=\) | \(\ds \set {1, 2, 5}\) | ||||||||||||
| \(\ds A_3\) | \(=\) | \(\ds \set {2, 4, 6, 8, 12}\) |
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:
| \(\ds A\) | \(=\) | \(\ds \set {1, i, -i}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {2, 1, -i}\) | ||||||||||||
| \(\ds C\) | \(=\) | \(\ds \set {i, -1, 1 + i}\) |
Then:
- $\paren {A \cup B} \cup C = \set {2, 1, -i,1, 1 + i}$
Example: People who are Blue-Eyed or British
Let:
| \(\ds B\) | \(=\) | \(\ds \set {\text {British people} }\) | ||||||||||||
| \(\ds C\) | \(=\) | \(\ds \set {\text {Blue-eyed people} }\) |
Then:
- $B \cup C = \set {\text {People who are blue-eyed or British or both} }$
Example: Overlapping Integer Sets
Let:
| \(\ds A\) | \(=\) | \(\ds \set {x \in \Z: 2 \le x}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {x \in \Z: x \le 5}\) |
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:
| \(\ds A\) | \(=\) | \(\ds \set {z \in \C: \cmod {z - 1} < 3}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {z \in \C: \cmod {z - 2 i} < 2}\) |
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:
| \(\ds A\) | \(=\) | \(\ds \set {z \in \C: \cmod {z + i} < 3}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {z \in \C: \cmod z < 5}\) | ||||||||||||
| \(\ds C\) | \(=\) | \(\ds \set {z \in \C: \cmod {z + 1} < 4}\) |
Then $A \cup B \cup C$ can be illustrated graphically as:
where the union is depicted in yellow.
Arbitrary Example $1$
Let:
| \(\ds A\) | \(=\) | \(\ds \set {1, 2}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {2, 3}\) |
Then:
- $A \cup B = \set {1, 2, 3}$
Arbitrary Example $2$
Let:
| \(\ds A\) | \(=\) | \(\ds \set {1, 2, 3, 4}\) | ||||||||||||
| \(\ds B\) | \(=\) | \(\ds \set {1, 4, 5, 6}\) |
Then:
- $A \cup B = \set {1, 2, 3, 4, 5, 6}$