Math 1010 On-line - Intervals

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Department of Mathematics --- College of Science --- University of Utah
Mathematics 1010 online

Intervals

Intervals are chunks of the real line. Specifically we define for real numbers $ a $ and $ b $:

  • $ [a,b] $: the set of all numbers $ x $ satisfying $ a \leq x \leq b $.
  • $ (a,b) $: the set of all numbers $ x $ satisfying $ a < x < b $.

  • $ [a,b) $: the set of all numbers $ x $ satisfying $ a \leq x < b $.

  • $ (a,b] $: the set of all numbers $ x $ satisfying $ a < x \leq b $.

So, for example, the interval $ [3,5) $ is the set of all numbers greater than or equal to $ 3 $ and less than $ 5 $. The numbers $ 3 $, $ 4 $, and $ 4.5 $ are in the interval, the numbers $ 5 $, $ 6 $ and $ -17 $ are not.

The notation may be a little confusing, but just remember that square brackets mean the end point is included, and round parentheses mean it's excluded. If both end points are included the interval is said to be closed, if they are both excluded it's said to be open. If one is included and the other excluded the interval is half open (or half closed, depending on your preference).

Now things get a little murky because the above notation is also used with $ b $ replaced with $ \infty $ or $ a $ replaced with $ -\infty $ (and only round parentheses at that end). This means that the interval is unlimited on the right or left, respectively.

For example the notation $ (1,\infty) $ is a fancy way of describing the set of all numbers greater than $ 1 $, and $ (-\infty,\pi] $ means the set of all number less than or equal to $ \pi $. The set of all real numbers can be expressed as $ (-\infty,\infty) $.

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