More Operations with Functions: Explanation

We typically describe subsets of the real number line using interval notation.  An interval is:

A set of all real numbers between two given endpoints that may or may not include one or both endpoints. 

We typically represent intervals in four different ways, with:

• plain English,

• tracing on a number line,

• inequalities, and

• interval notation.

All four methods start by specifying the two endpoints, but use different ways to indicate whether or not to include each endpoint.  Compare these four different methods:

English	Number line	Inequalities	Interval Notation
All numbers, x, between 1 and 2, not including either endpoint.		1 < x < 2	(1, 2)
All numbers, x, between 1 and 2, including 1 but not 2.		1 ≤ x <  2	[1, 2)
All numbers, x, between 1 and 2, including 2 but not 1.		1 < x ≤  2	(1, 2]
All numbers, x, between 1 and 2, including both endpoints.		1 ≤ x ≤  2	[1, 2]

In short, we have the correspondence:

While this covers the basics of interval notation, there are a few miscellaneous details that we should mention to avoid potential confusion.

• When using interval notation, it is permissible to reverse the direction of the inequality signs, as long as you do so consistently.  For example, "2 > x > 1" is ok, but "1 < x > 2" or "1 > x > 2" are not.  That is because this notation is actually short-hand for two separate inequalities: "1 < x < 2" means "1 < x and x < 2", that is "x is greater than 1 and less than 2"; this means that something like "1 > x > 2" would mean "x is less than 1 and greater than 2," which is not what we mean (in fact, it is impossible).

• For intervals that go on forever in one direction or the other, i.e., there is no end point, we invent two new "numbers", "-¥" (negative infinity) and "¥" (infinity) to serve as endpoints in interval notation.  We think of these as numbers that are, respectively, less than and greater than all real numbers.  For example, "(-¥, 0)" stands for "all numbers less than 0" or "x < 0", which we would draw as:

  

  Likewise, "[0, ¥)" stands for "all numbers greater than or equal to 0" or "0 ≤ x", which we would draw as:

  

  Note: The symbol ¥ for "infinity" was probably invented in 1656 by John Wallis (1616-1703); while Wallis discovered many of the important results of Calculus, it was his most famous student, Isaac Newton, to apply and popularize them.

• Finally, if you want to specify a region that is made up of several pieces, such as:

  

  this would be described as:

  English	Inequalities	Interval Notation
  x is greater than 1 and less than 2,  or greater than 3 and less than 4	1 < x < 2 or 3 < x < 4	(1, 2) È (3, 4)

   Notice that we must use the word "or" instead of the word "and".  You might naturally think that we are "adding" intervals together, so the word "and" would be more appropriate.  However, from a logical standpoint, if we used "and" then x would have to be less than 2 and greater than 3, which is impossible.  Also, from a set-theoretic standpoint, this is a union of the two intervals and the word "or" corresponds directly with the union sign "È".

Go back to More Operations with Functions

Table of Contents

Send questions or comments to jwicks@northpark.edu

Glossary