Section 1.6 Union
Definition 1.6.1.
Let \(S\) and \(T\) be two sets. The union \(S \cup T\) is the set consisting of the elements of either \(S\) or \(T\text{.}\) That is, \(x \in S \cup T\) exactly when either \(x \in S\) or \(x \in T\text{.}\) One can write
\begin{equation*}
S \cup T = \{x : (x \in S) \vee (x \in T)\}
\end{equation*}
When \(S\) and \(T\) are subsets of some common set \(U\) and are given by set-builder notation, we can use disjunction to describe the union. That is, if
\begin{equation*}
S = \{x \in U : \phi(x) \}
\end{equation*}
for a statement \(\phi(x)\text{,}\) and if
\begin{equation*}
T = \{x \in U : \psi(x) \}
\end{equation*}
for a statement \(\psi(x)\text{,}\) then the union is given by
\begin{equation*}
S \cup T = \{x \in U : \phi(x) \vee \psi(x)\}\text{.}
\end{equation*}
Example 1.6.2. Natural numbers divisible by either \(2\) or \(5\).
Consider the sets
\begin{equation*}
S = \{n \in \NN_0 : n\text{ is divisible by } 2\} \text{ and } T = \{n \in \NN_0 : n\text{ is divisible by } 5\}\text{.}
\end{equation*}
Then the union is
\begin{equation*}
S \cup T = \{n \in \NN_0 : n\text{ is divisible by either } 2\text{ or } 5\}\text{.}
\end{equation*}
We can write out a few of the elements:
\begin{equation*}
S \cup T =\{0,2,4,5,6,8,10,12,14,15,\dots\}\text{.}
\end{equation*}
Example 1.6.3. Natural numbers greater than \(5\) or less than \(7\).
Consider the sets
\begin{equation*}
S = \{x \in \NN_0 : x \geq 5\}
\end{equation*}
and
\begin{equation*}
T = \{x \in \NN_0 : x \leq 7\}\text{.}
\end{equation*}
Then the union is
\begin{equation*}
S \cup T = \{x \in \NN_0 : (x \geq 5) \vee (x \leq 7) \} = \NN_0\text{.}
\end{equation*}
Example 1.6.4. Natural numbers greater than \(7\) or less than \(5\).
Consider the sets
\begin{equation*}
S = \{x \in \NN_0 : x \geq 7\}
\end{equation*}
and
\begin{equation*}
T = \{x \in \NN_0 : x \leq 5\}\text{.}
\end{equation*}
Then the union is
\begin{equation*}
S \cup T = \{x \in \NN_0 : (x \geq 7)\vee(x \leq 5) \} = \{0,1,2,3,4,5,7,8,\dots\}\text{,}
\end{equation*}
the set of all natural numbers apart from \(6\text{.}\)