Section 1.1 continued
Set builder notation
Set builder notation is a way to construct sets out of other sets.
\[ A = \{x\in \mathbb{Z}: x\ge 0\} \mathrm{\ or\ } A = \{x : x\in\mathbb{Z}, x\ge 0\} \]
- A is the set of integers that are greater than or equal to zero
\[ E = \{2n : n\in\mathbb{Z}\} \]
- E is the set of things of the form \(2n\) where \(n\) is an integer
Set builder notation continued
More generally, set builder notation looks like this:
\[ X = \{\mathrm{expression} : \mathrm{rule}\} \]
and it captures all values of the expression that satisfy the rule.
Intervals of \(\mathbb{R}\)
Intervals are examples of sets given by set builder notation.
- \((a,b) = \{x\in\mathbb{R}: x>a \mathrm{\ and\ } x<b\}\) “open”
- \([a,b) = \{x\in\mathbb{R}: x\ge a \mathrm{\ and\ } x<b\}\) “half open”
- \((a,b] = \{x\in\mathbb{R}: x>a \mathrm{\ and\ } x\le b\}\) “half open”
- \([a,b] = \{x\in\mathbb{R}: x\ge a \mathrm{\ and\ } x\le b\}\) “closed”
- \([a,\infty) = \{x\in\mathbb{R}: x\ge a\}\) “infinite”
- \((a,\infty) = \{x\in\mathbb{R}: x>a\}\) “infinite”
- \((\infty,a) = \{x\in\mathbb{R}: x< a\}\) “infinite”
- \((\infty,a] = \{x\in\mathbb{R}: x\le a\}\) “infinite”