Section 1.4 Intervals on the real line
In what follows, we assume that \(x\) and \(y\) are real numbers.
Set builder notation is a way of describing sets of real numbers that satisfy some condition: \(\{ x:\) "some condition" \(\}\) describes the set of all real numbers for which the condition is true.
For example: \(\{ x : -2 \leq x \leq 5 \}\) represents the set of all real numbers that are greater than or equal to \(-2\) and less than or equal to \(5\text{.}\) This set is often described using interval notation; see below.
Interval Notation. Intervals of numbers on the real line are denoted using brackets. Different types of brackets have different meanings.
- \(\big[\) and \(\big]\) are used to indicate that the endpoints of the interval are included. These correspond to \(\leq\) and \(\geq\) when the interval is described using inequalities. For example, \(\big[ a, b \big] = \{ x: a\leq x\leq b\}\) . Therefore \(\big[ 0, 9 \big] = \{ x : 0 \leq x \leq 9 \}\text{.}\) This set contains numbers such as 0, 4, \(\pi\) , 8.3759838459, \(\frac{1}{6}\text{,}\) 9, etc. There are infinitely many members of this set.
- \(\big(\) and \(\big)\) are used to indicate that the endpoints of the interval are not included. These correspond to \(\lt\) and \(>\) when the interval is described using inequalities. For example, \(\big( a, b \big) = \{ x: a \lt x \lt b\}\) . Therefore, \(\big( 2, 5 \big)\) = \(\{ x : 2 < x < 5 \}\text{.}\) This set contains numbers such as 2.0000000001, 4, \(\pi\text{,}\) \(\frac{7}{3}\) etc. There are also infinitely many members of this set.
- The two types of brackets discussed above can be used together. For example, \(\big[ -3, 7 \big)\) = \(\{ x : -3 \leq x < 7 \}\text{.}\) This set also has infinitely members, and in particular contains \(-3\) but not 7.
- Note: Since infinity is not a real number, we must always use an open bracket for infinity or negative infinity. ex \(\big ( -\infty , 4 \big ]\text{.}\)
Combining intervals. The set \(\{x : 2 < x < 5 \mbox{ or } 6 < x < 9\}\) is the set of all real numbers that are in the interval \((2,5)\) or are in the interval \((6,9)\text{.}\) We use the \(\cup\) symbol to signify the union of two sets. For example, \(\{x : 2 < x < 5 \mbox{ or } 6 < x < 9\}=(2,5)\cup(6,9)\text{,}\) and
\(\{x : x\neq -4 \} = (-\infty,-4)\cup(-4,\infty)\text{.}\) More generally, if \(A\) and \(B\) are two sets of real numbers then their union, denoted \(A \cup B\text{,}\) is the set of all numbers that are in \(A\text{,}\) \(B\text{,}\) or both. Their intersection, denoted \(A \cap B\text{,}\) is the set of all numbers that are in both \(A\) and \(B\text{.}\)
For example, \((-\infty, 2) \cap [-1, 5]\) is the set of all real numbers \(x\) that are less than \(2\) and satisfy \(-1\leq x \leq 5\text{.}\) That is, \((-\infty, 2) \cap [-1,5]=[-1,2)\text{.}\) Another way to think about this set is as the solution set to the system of inequalities (or compound inequality) \(x \lt 2,\ \ x \geq -1,\ \ x \leq 5\text{.}\)
Exercises Practice Problems
Write the following expressions in set builder notation or interval notation:
1.
\(\big[ -1, 10 \big]\)
Answer.
\(\{ x : -1 \leq x \leq 10 \}\)
2.
\(\{ x : -7 \leq x < 5 \}\)
Answer.
\(\big[ -7, 5 \big)\)
3.
\(\big[ 7, 28 \big)\)
Answer.
\(\{ x : 7 \leq x \lt 28 \}\)
4.
\(\{ x : 2 \lt x \lt 7 \}\)
Answer.
\(\big( 2, 7 \big)\)
5.
\(\big( -8, 5 \big)\)
Answer.
\(\{ x : -8 \lt x \lt 5 \}\)
6.
\(\{ x : 0 \lt x \leq 6 \}\)
Answer.
\(\big( 0, 6 \big]\)
7.
\(\{x : x > -3\}\)
Answer.
\((-3,\infty)\)
8.
\((-\infty,2)\cup(2,8)\cup(8,\infty)\)
Answer.
\(\{x : x\neq 2 \mbox{ and } x\neq 8\}\)
