Definition 5.6.
The contents of a set can be described by listing the elements of the set, separated by commas, inside a set of curly brackets. This way of describing a set is called roster form.
Section 5.2 Roster Form
So far we specified the elements of sets by verbally. The roster form introduced here offers a concise way of writing down sets by listing all elements of the set. Furthermore we use ellipsis to describe the elements in a set, when we believe that the reader understands how a pattern in a list of elements continues.
In the video in Figure 5.7 we recall the definition of roster form and give first examples.
We give examples of sets in roster form along with a verbal description.
Example 5.8. Sets in roster form.
- \(\{1, 2, 3, 4\}\) is the set containing the numbers 1, 2, 3, and 4.
- \(\{\mathtt{w}, \mathtt{x}, \mathtt{y}, \mathtt{z}\}\) is the set containing the letters \(\mathtt{w}, \mathtt{x}, \mathtt{y}\text{,}\) and \(\mathtt{z}\text{.}\)
- \(\{ \text{red}, \text{yellow}, \text{blue} \}\) is the set containing red, yellow, and blue.
- \(\{6\}\) is the set containing the number \(6\text{.}\)
- \(\{3,-3,11\}\) is the set containing the numbers \(3\text{,}\) \(-3\text{,}\) and \(11\text{.}\)
- \(\{5,3,\mathtt{w}\}\) is the set containing the numbers \(5\) and \(3\) and the letter \(\mathtt{w}\text{.}\)
Now try yourself to translate a verbal description of a set into roster form.
Checkpoint 5.9. Write the set in roster form.
Give the set of natural numbers less than \(7\) in roster form:
\(\lbrace\)\(\rbrace\)
Recall that an ellipsis (\(\ldots\)) indicates that the pattern is continued. We can use an ellipsis when writing a set in roster form instead of listing every element.
Example 5.10. Sets in roster from with ellipsis.
We give examples of sets written in roster form that use ellipses.
- \(\{1,2,3,\ldots,100\}\) is the set of integers from \(1\) to \(100\text{.}\)
- \(\{2,3,4,\ldots,99\}\) is the set of integers from \(2\) to \(99\text{.}\)
- \(\{\mathtt{c},\mathtt{d},\mathtt{e},\ldots,\mathtt{n}\}\) is the set of letters from \(\mathtt{c}\) to \(\mathtt{n}\text{.}\)
Convert a set given in roster form with ellipsis into roster form without ellipsis.
Checkpoint 5.11. Write the set in roster form.
Give the set in roster form (without ellipsis):
\(\lbrace8,9,10,...,16\rbrace =
\lbrace\)\(\rbrace\)
Roster form also allows us to formulate a set that does not contain any elements by writing \(\{ \}\text{.}\)
Definition 5.12.
The empty set (also called the null set) is the set that consists of no elements. It is denoted by \(\{ \}\text{.}\)
The empty set contains no elements. In particular it does not contain the number \(0\text{.}\)
