The Real Number Line, Interval Notation and Set Notation

In this section, you will learn about three different ways in which to write down sets of solutions.

The Real Number Line.

The most intuitive way is to use the real number line. If we draw a line, designate a point on the line to be zero, and choose a scale, then every point on the line corresponds uniquely to a real number, and vice versa:

The real number line "respects" the order of the real numbers. A bigger number will always be found to the right of a smaller number. In the picture below, a>b.

We visualize a set on the real number line by marking its members.

It is standard to agree on the following conventions: To include an endpoint, we "bubble it in." To exclude an endpoint, we use an "empty bubble". Here is the set of all real numbers greater than -2 and less than or equal to 5:

The number -2 is excluded from the set, so you see an "empty bubble"; the number 5 is included in the set, so the bubble at 5 is "filled in."

Next comes an unbounded set, the set of all numbers less than or equal to 3:

The set does not need to be "connected." The following graph depicts all real numbers which are either greater than 2 or strictly between -1 and 1.

The following is a description of the set of all real numbers with the exception of -1 and 2:

Interval Notation.

Interval notation translates the information from the real number line into symbols.

Our example

becomes the interval (-2,5].

To indicate that an endpoint is included, we use a square bracket; to exclude an endpoint, we use parentheses.

Our example

is written in interval notation as tex2html_wrap_inline125 . The infinity symbols " tex2html_wrap_inline127 " and " tex2html_wrap_inline129 " are used to indicate that the set is unbounded in the positive ( tex2html_wrap_inline127 ) or negative ( tex2html_wrap_inline129 ) direction of the real number line. " tex2html_wrap_inline127 " and " tex2html_wrap_inline129 " are not real numbers, just symbols. Therefore we always exclude them as endpoints by using parentheses.

If the set consists of several disconnected pieces, we use the symbol for union " tex2html_wrap_inline139 ":

Our example

is written in interval notation as tex2html_wrap_inline141 .

How could we write down

in interval notation? There are three pieces to consider:

displaymath121

An interval such as tex2html_wrap_inline143 , where both endpoints are excluded is called an open interval. An interval is called closed, if it contains its endpoints, such as tex2html_wrap_inline145 .

An unbounded interval such as tex2html_wrap_inline147 is considered to be open; an interval such as tex2html_wrap_inline149 is called closed (even though it does not contain its right endpoint). The whole real line tex2html_wrap_inline151 is considered to be both open and closed. (So intervals are not like doors, they can be open and closed at the same time.)

Set Notation.

The most flexible (and complicated?) way to write down sets is to use set notation.

Sets are delimited by curly braces. You can write down finite sets as lists.

For instance

displaymath153

is the set with the three elements -1, tex2html_wrap_inline171 and tex2html_wrap_inline173 .

For sets with infinitely many elements this becomes impossible, so there are other ways to write them down.

Special symbols are used to denote important sets:

Beyond that, set notation uses descriptions: the interval (-3,5] is written in set notation as

displaymath154

read as " the set of all real numbers x such that tex2html_wrap_inline181 ."

The first part tells us what "universe" of numbers we are considering (in our case the universe of real numbers), the delimiter " tex2html_wrap_inline183 " separates the "universe" part from the second part, where we describe the property our numbers in the set are supposed to satisfy.

The set

displaymath155

is the set of all integers exceeding -3 and not greater than 5; this is a finite set; we could write it as a list,

displaymath156

The set

displaymath157

is even smaller; it contains only five elements:

displaymath158

Here are some more examples:

The interval tex2html_wrap_inline185 can be written as

displaymath159

the set tex2html_wrap_inline187 looks like this in set notation:

displaymath160

or like this

displaymath161

Exercise 1.

Write the set of all real numbers strictly between -2 and tex2html_wrap_inline171 in interval notation and in set notation.

Answer.

Exercise 2.

Write the set tex2html_wrap_inline195 in set notation.

Answer.

Exercise 3.

Write the unbounded set

in both interval notation and set notation.

Answer.

Exercise 4.

Mark the set

displaymath203

on the real number line.

Answer.

Exercise 5.

Write down the set of solutions to the inequality

displaymath205

in all three notations.

Answer.

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