Section 1.4 Intersection
Definition 1.4.1.
Let \(S\) and \(T\) be two sets. The intersection \(S \cap T\) is the set consisting of the elements \(S\) and \(T\) have in common. That is, \(x \in S \cap T\) exactly when both \(x \in S\) and \(x \in T\text{.}\) One can write
\begin{equation*}
S \cap T = \{x : (x \in S) \wedge (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 conjunction to describe the intersection. That is, if
\begin{equation*}
S = \{x \in U : \phi(x) \}
\end{equation*}
for a statement \(\phi(x)\) with free variable \(x\text{,}\) and if
\begin{equation*}
T = \{x \in U : \psi(x) \}
\end{equation*}
for a statement \(\psi(x)\text{,}\) then the intersection is given by
\begin{equation*}
S \cap T = \{x \in U : \phi(x) \wedge \psi(x)\}\text{.}
\end{equation*}
Example 1.4.2. Natural numbers divisible by \(10\).
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 intersection is
\begin{equation*}
S \cap T = \{n \in \NN_0 : n\text{ is divisible by both } 2\text{ and } 5\}\text{.}
\end{equation*}
From this you can see that:
\begin{equation*}
S \cap T = \{n \in \NN_0 : n\text{ is divisible by } 10\}=\{0,10,20,\dots\}\text{.}
\end{equation*}
Example 1.4.3. Natural numbers between \(5\) and \(7\).
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 intersection is
\begin{equation*}
S \cap T = \{x \in \NN_0 : 5 \leq x \leq 7 \} = \{5,6,7\}\text{.}
\end{equation*}
Example 1.4.4. Natural numbers between \(7\) and \(5\).
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 intersection is
\begin{equation*}
S \cap T = \{x \in \NN_0 : (x \geq 7)\wedge(x \leq 5) \} = \varnothing\text{.}
\end{equation*}
We have a word for this situation.
Definition 1.4.5.
Let \(U\) be a set, and let \(S \subseteq U\) and \(T\subseteq U\) be subsets. We say that \(S\) and \(T\) are disjoint if and only if \(S \cap T = \varnothing \text{.}\)