We have discussed the notion of set and of subsets. In calculus, the focus is on 'functions' between sets of real numbers. Later, we will discuss the concept of functions. Here, we will focus on sets of real numbers. A set of real numbers that contains all real numbers lying between any two numbers of the set is called a bounded interval.
Consider the bounded interval of all real numbers between #1# and #5#. It is intuitive that the numbers #\pi# and #4# are in this interval. But, what about the numbers #1# or #5#; are they also included? In order to provide a precise specification, we distinguish four kinds of bounded interval.
Let #\blue c#, #\green d# be two real numbers with #\blue c \le \green d#. We define the following four types of bounded intervals.
| Name interval |
Interval notation |
Set-builder notation |
Example |
| open |
#\ivoo{\blue c}{\green d}# |
#\{x\in\mathbb{R}\mid \blue c\lt x\lt \green d\}# |
#\ivoo{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \lt x\lt \green 8\}# |
| closed |
#\ivcc{\blue c}{\green d}# |
#\{x\in\mathbb{R}\mid \blue c\le x\le \green d\}# |
#\ivcc{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \le x\le \green 8\}# |
| open-closed |
#\ivoc{\blue c}{\green d}# |
#\{x\in\mathbb{R}\mid \blue c\lt x\le \green d\}# |
#\ivoc{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \lt x\le \green 8\}# |
| closed-open |
#\ivco{\blue c}{\green d}# |
#\{x\in\mathbb{R}\mid \blue c\le x\lt \green d\}# |
#\ivco{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \le x\lt \green 8\}# |
The point #\blue c# is called the left boundary or left end point and the point #\green d# the right boundary or right end point of the interval. The points #\blue c# and #\green d# are the end points or boundary points of the interval.
Points #x\in\mathbb{R}# with #\blue c\lt x\lt \green d# are called interior points of the interval.
The length of the interval is #\left|\blue c- \green d\right|#.
The set #\mathbb{R}# of real numbers can be visualized as the real number line. Elements of a line are often called points. This explains why we speak of real numbers as points of the real line.
Every real number can be associated with a single point on the real number line. As a result an interval can be represented by a segment of the real number line.
Example
#{}#
#\ivoo{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \lt x\lt \green 8\}#
#{}#
#\ivcc{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \le x\le \green 8\}#
#{}#
#\ivco{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \le x\lt \green 8\}#
#{}#
#\ivoc{\blue 3}{\green 8} = \{x\in\mathbb{R}\mid \blue 3 \lt x\le \green 8\}#
On the segment of the real number line, we mark both real number with a dot and if there is a parenthesis the dot is open (empty), while if there is a square bracket the dot is closed.
In logic, the operator #\land# denotes "and", and a chain of inequalities like # \blue c\le x\le \green d# is an abbreviation of #x \ge \blue c \text{ and } x \le \green d#, so we can write the interval #\ivcc{\blue c}{\green d}# as \[\{x\in\mathbb{R}\mid \blue c\le x\le \green d\}\quad \text{ or } \quad\{x\in\mathbb{R}\mid x \ge \blue c \land x \le \green d\}\]
If #\blue c=\green d#, then this set consists of the single point #\blue c# and if #\blue c\gt \green d#, then it is the empty set.
For example,
- #\{x\in\mathbb{R}\mid x \ge \blue{-2} \land x \le \green{5} \} = \ivcc{\blue{-2}}{\green{5}}#
- #\{x\in\mathbb{R}\mid x \ge \blue{-3} \land x \le \green{-3} \} = \{\blue{-3}\}#
- #\{x\in\mathbb{R}\mid x \ge \blue{3} \land x \le \green{-3} \} = \emptyset#
A bounded interval is an interval for which the end points are real numbers. Bounded intervals are also known as finite intervals. The adjective finite means that the length of the interval is finite; it does not mean that the interval is finite.
Actually the only intervals that are finite sets are of the form #\ivcc{\blue c}{\blue c}# for some real number #\blue c#. So a more precise description of a bounded interval is interval of finite length.
There also are unbounded intervals. In these case there is at most one real end point. If there is no left end point, then we designate #-\infty# as the left end point, which is not a real number. Similarly, if there is no right end point, we use the notation #\infty# to replace the missing the right end point.
Let #\blue c#, #\green d# be two real numbers. We define the following unbounded interval
| Name interval |
Interval notation |
Set-builder notation |
Example |
| left-open |
#\ivoo{\blue c}{\infty}# |
#\{x\in\mathbb{R}\mid x \gt \blue c \}# |
#\ivoo{\blue 3}{\infty} = \{x\in\mathbb{R}\mid x\gt \blue 3\}# |
| left-closed |
#\ivco{\blue c}{\infty}# |
#\{x\in\mathbb{R}\mid x \ge \blue c \}# |
#\ivco{\blue 3}{\infty} = \{x\in\mathbb{R}\mid x\ge \blue 3\}# |
| right-closed |
#\ivoc{-\infty}{\green d}# |
#\{x\in\mathbb{R}\mid x \le \green d\}# |
#\ivoc{-\infty}{\green 8} = \{x\in\mathbb{R}\mid x \le \green 8\}# |
| right-open |
#\ivoo{-\infty}{\green d}# |
#\{x\in\mathbb{R}\mid x\lt \green d\}# |
#\ivoo{-\infty}{\green 8} = \{x\in\mathbb{R}\mid x \lt \green 8\}# |
The interval #\ivoo{-\infty}{\infty}# is unbounded at both ends (simultaneously open and closed). It is in fact the set of all real numbers #\mathbb{R}#. Remember that the bracket adjacent to an infinity sign is always a parenthesis.
The symbols #\infty# and #-\infty# represent "infinity" and "minus infinity".
Intuitively infinity should be larger than any real number, while minus infinity should be smaller than any real number. Using our intuition we will allow for inequalities involving symbols #\infty# and #-\infty#.
The convention is that #-\infty \lt x# and #x\lt \infty# for each real number #x#. Thus, \[\{x\in\mathbb{R}\mid x\le \green d\}=\{x\in\mathbb{R}\mid -\infty\lt x\le \green d\}\]
An unbounded interval can also be represented by a segment of the real number line. We use an infinite end point to indicate that there is no bound in the positive direction, while for minus infinity there is no bound in the negative direction.
Example
#{}#
#~\ivoo{\blue 3}{\infty}~ = \{x\in\mathbb{R}\mid x\ge \blue 3\}#
#{}#
#~\ivco{\blue 3}{\infty}~ = \{x\in\mathbb{R}\mid x\gt \blue 3\}#
#{}#
#\ivoo{-\infty}{\green 8} = \{x\in\mathbb{R}\mid x \lt \green 8\}#
#{}#
#\ivoc{-\infty}{\green 8} = \{x\in\mathbb{R} \mid x \le \green 8\}#
The logic operator #\lor# denotes "#or#". Thus, the set #\{x\in\mathbb{R}\mid x \le \green d \text{ or } x \gt \blue c\}# can also be written as #\{x\in\mathbb{R}\mid x \le \green d \lor x \gt \blue c\}#.
If #{\blue c}\le {\green d}#, then this set is the whole real line #\mathbb{R}#.
Write the set # \{x\in\mathbb{R}\mid 9 \le x \le 12\}# in the interval notation.
#\ivcc{9}{12}#
The interval notation of #\{x\in\mathbb{R}\mid 9 \le x \le 12\}# is #\ivcc{9}{12}#.
Intervals
