Set Union/Examples

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Examples of Set Union

Example: $2$ Arbitrarily Chosen Sets

Let:

\(\displaystyle S\) \(=\) \(\displaystyle \set {a, b, c}\) $\quad$ $\quad$
\(\displaystyle T\) \(=\) \(\displaystyle \set {c, e, f, b}\) $\quad$ $\quad$

Then:

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


Example: $2$ Arbitrarily Chosen Sets of Complex Numbers

Let:

\(\displaystyle A\) \(=\) \(\displaystyle \set {3, -i, 4, 2 + i, 5}\) $\quad$ $\quad$
\(\displaystyle B\) \(=\) \(\displaystyle \set {-i, 0, -1, 2 + i}\) $\quad$ $\quad$

Then:

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


Example: $3$ Arbitrarily Chosen Sets

Let:

\(\displaystyle A_1\) \(=\) \(\displaystyle \set {1, 2, 3, 4}\) $\quad$ $\quad$
\(\displaystyle A_2\) \(=\) \(\displaystyle \set {1, 2, 5}\) $\quad$ $\quad$
\(\displaystyle A_3\) \(=\) \(\displaystyle \set {2, 4, 6, 8, 12}\) $\quad$ $\quad$

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:

\(\displaystyle A\) \(=\) \(\displaystyle \set {1, i, -i}\) $\quad$ $\quad$
\(\displaystyle B\) \(=\) \(\displaystyle \set {2, 1, -i}\) $\quad$ $\quad$
\(\displaystyle C\) \(=\) \(\displaystyle \set {i, -1, 1 + i}\) $\quad$ $\quad$

Then:

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


Example: People who are Blue-Eyed or British

Let:

\(\displaystyle B\) \(=\) \(\displaystyle \set {\text {British people} }\) $\quad$ $\quad$
\(\displaystyle C\) \(=\) \(\displaystyle \set {\text {Blue-eyed people} }\) $\quad$ $\quad$

Then:

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


Example: Overlapping Integer Sets

Let:

\(\displaystyle A\) \(=\) \(\displaystyle \set {x \in \Z: 2 \le x}\) $\quad$ $\quad$
\(\displaystyle B\) \(=\) \(\displaystyle \set {x \in \Z: x \le 5}\) $\quad$ $\quad$

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:

\(\displaystyle A\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z - 1} < 3}\) $\quad$ $\quad$
\(\displaystyle B\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z - 2 i} < 2}\) $\quad$ $\quad$

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

Set-Union-Circles-in-Complex-Plane.png

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:

\(\displaystyle A\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z + i} < 3}\) $\quad$ $\quad$
\(\displaystyle B\) \(=\) \(\displaystyle \set {z \in \C: \cmod z < 5}\) $\quad$ $\quad$
\(\displaystyle C\) \(=\) \(\displaystyle \set {z \in \C: \cmod {z + 1} < 4}\) $\quad$ $\quad$

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

Set-Union-3-Circles-in-Complex-Plane.png

where the union is depicted in yellow.