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Section 5.1 Definition of a Set
A set is a collection of objects. But not every description of a collection has the same meaning for everyone.
Speakers of different languages may understand “the last four letters of the alphabet” in different ways. A clearer name for the collection of the letters
\(\mathtt{w}, \mathtt{x}, \mathtt{y}\text{,}\) and
\(\mathtt{z}\) is “the last four letters of the English alphabet.”
The video in
Figure 5.1.1 introduces sets. A more detailed discussion follows.
Figure 5.1.1. Definition of a Set by Matt Farmer and Stephen Steward When a description is not ambiguous, we say it is well-defined. We also call a collection of objects
well-defined if its contents can be clearly identified.
Example 5.1.2 . Collections of objects.
We give descriptions of collections of objects that are well-defined or not well-defined.
The collection of Greek letters is well-defined.
The collection of South American countries is well-defined.
The collection of cute animals is not well-defined. The characteristics that make an animal “cute” are a matter of opinion.
The collection of best math teachers is not well-defined. The meaning of the word “best” here is up for interpretation.
Now that we understand a well-defined collection, we can give the formal definition of a set.
Definition 5.1.3 .
A
set is a well-defined collection of distinct objects. The objects in a set are called
elements of the set.
When we use variables as placeholders for sets, we often use capital letters such as
\(A\) or
\(B\text{.}\) Sets may be described in several ways. For example, the set consisting of the letters
\(\mathtt{w}, \mathtt{x}, \mathtt{y}\text{,}\) and
\(\mathtt{z}\) might also be described as the set consisting of the last four letters of the English alphabet. So far, we have shown sets by giving a verbal description of the contents of the set. Two more methods we will use to show a set are
roster form and
set-builder notation .
