# Set Union/Examples

## 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: $3$ 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: 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:

$W \subseteq U \cup V$