Example 1.62. Finding a Union.
The union of \([-3,1]\) and \((0,+\infty)\) 
results in the interval \([-3,+\infty)\text{.}\) 
This is written as
\begin{equation*}
[-3,1] \cup (0,+\infty) = [-3,+\infty).
\end{equation*}
Section 1.11 Unions and Intersections
We may combine two intervals in two different ways: by forming their union or intersection. First, the union of two intervals consists of all real numbers that are in one OR both of the intervals. Graphically, this combines the two intervals into a (usually) larger set which may or may not be an interval. We use the symbol “\(\cup\)” to denote the union of two sets.
The intersection of two intervals consists of all real numbers that are in both of the intervals. It is the set of all real numbers they have in common. The symbol for intersection is “\(\cap\)”.
Example 1.63. Finding an Intersection.
The intersection of \([-3,1]\) and \((0,+\infty)\) is \((0,1]\) 
and this result is written as
\begin{equation*}
[-3,1] \cap (0,+\infty) = (0,1].
\end{equation*}
If two sets have nothing in common, their intersection is the empty set, that is the set containing no real numbers. This is denoted by the symbol \(\emptyset\text{.}\)
Example 1.64.
The interval of \((-\infty,-3]\) and \((0,1]\) do not overlap. 
So their intersection is empty:
\begin{equation*}
(-\infty,-3] \cap (0,1] = \emptyset.
\end{equation*}
Remark 1.65.
The intersection or union of two intervals does not itself need to be an interval. For instance,
\begin{equation*}
(-\infty,-3)\cup (0,1]
\end{equation*}
is just comprised of real numbers in one of the two intervals above and cannot be written more simply than this.
Checkpoint 1.66.
Find the intersection of the non-negative real numbers and the interval \((-1,1)\text{.}\)
