We introduce sets as collections of objects. However, not all descriptions of a collection of objects are necessarily interpreted in the same way by everyone.
For example, “the last four letters of the alphabet” could be interpreted differently by speakers of different languages. So, a more precise way to describe the collection of the letters \(\mathtt{w}, \mathtt{x}, \mathtt{y}\text{,}\) and \(\mathtt{z}\) might be “the last four letters of the English alphabet.”
We give an introduction to sets in the video in Figure 5.1. It is followed by a more detailed discussion.
Figure5.1.Definition of a Set by Matt Farmer and Stephen Steward
When there is no such ambiguity in the description of an object, then we say it is well-defined. We extend this to collections of objects and call them well-defined if their contents can be clearly determined.
Example5.2.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 have some intuition what a well-defined collection is, we are ready to give the formal definition of a set.
Definition5.3.
A set is a well-defined collection of distinct objects. The objects in a set are called elements of the set.
Read the definition of the set again and then complete the following.
Checkpoint5.4.Definition of a Set.
Complete the definitions:
A set is a well-defined
select
flock
pile
bucket
heap
collection
stack
pack
list
of distinct
select
birds
bears
numbers
letters
words
objects
.
The
select
birds
numbers
objects
letters
words
wolves
in a set are called
select
things
elements
animals
characters
of the set.
Decide whether certain collections are sets.
Checkpoint5.5.Are these sets ?
Determine which of the following are sets:
The collection of funny XKCD comics.
The collection of difficult problems.
The collection of cuddly animals.
The collection of students in all sections of MAT 112 this semester.
When we use variables as placeholders for sets, we often use capital letters such as \(A\) or \(B\text{.}\) Sets may be described in various 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 indicated sets by giving a verbal description of the contents of the set. Two additional methods we will use to indicate a set are roster form and set-builder notation.
