Section1.7Notation for Intervals
Ā¶If you say
\begin{equation*} (\text{age of a voter})\geq18 \end{equation*}and have a particular voter in mind, what is that person's age? There's no way to know for sure. Maybe they are \(18\text{,}\) but maybe they are older. It's helpful to visualize the possibilities with a number line, as in FigureĀ 1.7.1.
The shaded portion of the number line in FigureĀ 1.7.1 is a mathematical interval. For now, that just means a collection of certain numbers. In this case, it's all the numbers \(18\) and above.
It's one thing to say \((\text{age of a voter})\geq18\text{,}\) and another thing to discuss all the shaded numbers in the interval in FigureĀ 1.7.1. In mathematics,
\begin{equation*} (\text{age of a voter})\geq18 \end{equation*}is saying that there is one age under consideration and all we know is that it's 18 or larger. It's subtle, but this is not the same thing as the collection of all numbers that are \(18\) or larger. Mathematics has two common ways to write down these kinds of collections.
Definition1.7.2Set-Builder Notation
Set-builder notation attempts to directly say the condition that numbers in the interval satisfy. In general, write set-builder notation like:
\begin{equation*} \left\{x\mid\text{condition on }x\right\} \end{equation*}and read it out loud as āthe set of all \(x\) such that ā¦.ā For example,
\begin{equation*} \left\{x\mid x\geq18\right\} \end{equation*}is read out loud as āthe set of all \(x\) such that \(x\) is greater than or equal to \(18\text{.}\)ā The breakdown is as follows.
| \(\highlight{\{}\lowlight{x\mid x\geq18}\highlight{\}}\) | the set of |
| \(\lowlight{\{}\highlight{x}\lowlight{{}\mid x\geq18\}}\) | all \(x\) |
| \(\lowlight{\{x}\highlight{{}\mid{}}\lowlight{x\geq18\}}\) | such that |
| \(\lowlight{\{x\mid{}}\highlight{x\geq18}\lowlight{\}}\) | \(x\) is greater than or equal to \(18\) |
Definition1.7.3Interval Notation
Interval notation tries to just say the numbers where the interval starts and stops. For example, in FigureĀ 1.7.1, the interval starts at \(18\text{.}\) To the right, the interval extends forever and has no end, so we use the \(\infty\) symbol (meaning "infinity"). This particular interval is denoted:
\begin{equation*} [18,\infty) \end{equation*}Why use ā\([\)ā on one side and ā\()\)ā on the other? The square bracket is telling you that \(18\) is part of the interval and the round parenthesis is telling you that \(\infty\) is not part of the interval.ā1āAnd how could it be, since \(\infty\) is not even a number?
In general there are four types of intervals. Take note of the different uses of round parentheses and square brackets.
Also we allow \(a\) or \(b\) to be the symbols \(\infty\) or \(-\infty\text{.}\) If these symbols are used then the interval extends forever in one direction. Wherever these symbols are used, there has to be a round parenthesis, not a square bracket, since the interval won't actually include these (non-) numbers.
Example1.7.8Lifespan
The person with the oldest verified age at the time of her death was Jeanne Calment of France, who died in 1997 at the precise age of \(122.45\) years old.
Consider a random human being from history. How many years did that person live? We can't be specific if we don't know who we are talking about. But we can give a range of possibilities. If a person was ever alive, they were older than \(0\) when they died. And using Jeanne Calment as the upper limit, no one has ever lived beyond \(122.45\) years. So FigureĀ 1.7.9 gives a picture of these possible ages.
This is an interval that starts at \(0\) (not including \(0\)) and ends at \(122.45\) (including \(122.45\)). So in interval notation, it's:
\begin{equation*} (0,122.45] \end{equation*}It's an interval where all the numbers are greater than \(0\) and less than or equal to \(122.45\text{.}\) So using set-builder notation, it's:
\begin{equation*} \left\{x\mid x\gt0\text{ and }x\leq122.45\right\} \end{equation*}Sometimes people like to combine those last two inequalities and write this like:
\begin{equation*} \left\{x\mid0\lt x\leq122.45\right\} \end{equation*}Exercise1.7.10Construction Recycling
Exercise1.7.11Interval and Set-Builder Notation from Number Lines
Occasionally there is a need to consider number line pictures such as FigureĀ 1.7.12, where two or more intervals appear.
This picture is trying to tell you to consider numbers that are between \(-5\) and \(1\text{,}\) together with numbers that are between \(4\) and \(7\text{.}\) That word ātogetherā is related to the word āunion,ā and in math the union symbol, \(\cup\text{,}\) captures this idea. So we can write the numbers in this picture as either
\begin{equation*} [-5,1]\cup(4,7] \end{equation*}(which uses interval notation) or as the much more tedious
\begin{equation*} \{x\mid-5\leq x\leq1\}\cup\{x\mid 4\lt x\leq7\} \end{equation*}(which uses set-builder notation). There is at least one more way to use set-builder notation here. If you recognize that the shaded numbers are either between \(-5\) and \(1\) or are between \(4\) and \(7\text{,}\) then you see that you may write:
\begin{equation*} \{x\mid-5\leq x\leq1\text{ or }4\lt x\leq7\} \end{equation*}Unions and the word āorā
In mathematics, the union of two things is often associated with the word āorā instead of the word āand.ā The reason for this is that to be part of a union, something is either in one of the sets or it is in the other. To be inside one set and inside the other set means you are inside both sets at the same time, and that is impossible with the two sets illustrated in FigureĀ 1.7.12. So while the picture of a union looks like one set and another set joined together, mathematicians will often interpret a picture like this as some number being part of one set or the other set.