9.2: Defining sets
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Remember that mathematical notation is about communicating mathematical information. Since a set is defined by its member objects, to communicate the details of a set of objects one needs to provide a means to decide whether any given object is or is not an element of the set.
9.2.1: Listing elements
One way to communicate the details of a set definition is to explicitly list or describe all elements of the set. Such a list should be enclosed in braces to indicate that the objects in the list are being collected into a set.
If we write
\begin{equation*} A = \{ \text{monkey, tennis ball, the number } 2 \} \text{,} \end{equation*}
then we intend for the letter \(A\) to become a label representing the set consisting of some specific monkey, some specific tennis ball, and the number \(2\text{.}\)
Here are some sets containing familiar collections of numbers. Notice how in the first two examples we “list” the elements by providing a pattern and then using … to imply that the pattern continues as expected, and in the second two examples we merely describe what the elements are in words.
the set \(\{ 0, 1, 2, \ldots \}\) of natural numbers
the set \(\{ \ldots , -2, -1, 0, 1, 2, \ldots \}\) of integers
the set of all fractions, call the set of rational numbers
the set of all decimal numbers, called the set of real numbers
Keep the following in mind for a set defined by listing elements.
- Order does not matter. For example, \(\{a,b\}\) and \(\{b,a\}\) are the same set because they consist of precisely the same member elements.
- Repetition does not matter. For example, \(\{a,a,b\}\) and \(\{a,b\}\) are the same set because they consist of precisely the same member elements.
9.2.2: Candidate-condition notation
Another way to define a set is candidate-condition notation:
This notation provides a means to decide whether an object is a member of the set by first using an already-defined set as a pool of “member candidates” as well a condition or a list of conditions each candidate must satisfy in order to actually be a member.
If we write \(S\) for the set being defined, \(C\) for the set of candidates, and \(T\) for the test those candidates must satisfy to be included in \(S\) (that is, \(T\) is a predicate with domain \(C\)), then the candidate-condition notation takes the form
and can be read as
\(S\) is the set of those elements \(x\) in \(C\) for which \(T(x)\) is true.
Consider the set
\begin{equation*} A = \{ 0, 3, 6, 9, 12, \ldots \} \text{.} \end{equation*}
We could define this set in a more precise manner (i.e. without resorting to using dots) as follows.
\begin{equation*} A = \{ n \in \mathbb{N} \vert n \text{ divisible by } 3 \} \end{equation*}
The “\(n\in\mathbb{N}\)” part to the left of the divider tells us that the pool of “member candidates” for \(A\) is the set of natural numbers, and the test to the right of the divider tells us how to decide when a given candidate natural number \(n\) is actually a member of \(A\text{.}\) In words, you should think of the above definition as saying the following.
Set \(A\) consists of those elements of \(\mathbb{N}\) which are divisible by \(3\text{.}\)
9.2.3: Form-parameter notation
Finally, sets can be defined by form-parameter notation:
This notation describes the members of a set by providing a “form” to which the members must conform. Usually the “form” is based on parameter variables that can range over a set of possibilities.
Again consider the set
\begin{equation*} A = \{ 0, 3, 6, 9, 12, \ldots \} \text{.} \end{equation*}
We could also define this set as
\begin{equation*} A = \{3n \vert n\in \mathbb{N} \} \text{.} \end{equation*}
Here, the form of the elements of \(A\) is given to the left of the divider as “\(3\) times a number”, where the number is represented by the parameter \(n\text{.}\) Then the allowed range of the number parameter \(n\) is given to the right of the divider. In words, you should think of the above definition as saying the following.
The elements of set \(A\) are precisely those objects that are \(3\) times a natural number.
We could define the set \(\mathbb{Q}\) of rational numbers in this way:
\begin{equation*} \mathbb{Q} =\{ \dfrac{m}{n} \vert m,n\in \mathbb{Z},\, n\ne 0\} \text{.} \end{equation*}
This says that the set \(\mathbb{Q}\) consists of all symbols of the form “number over number”, where the numbers can be any integers, as long as the bottom number is not zero. However, we need to be a little bit careful here, since we allow different symbols of this form to represent the same element. For example,
\begin{align*} \dfrac{3}{6} & = \dfrac{1}{2} \text{,} & \dfrac{2}{-9} & = \dfrac{-2}{9} \text{,} & \dfrac{0}{n} & = \dfrac{0}{1} \quad \text{(any } n\ne 0\text{).} \end{align*}
We really should make this element form duplication explicit in the definition of the set, but to do this would be really cumbersome and would be expressing something that is learned in grade school, so it is usually omitted.
9.2.4: Empty set
There is one special set, the elements of which are very easy to list.
the set which has no elements
the empty set
The empty set is defined by requiring that the statement “\(x\) is an element of \(\emptyset\)” is always false, for every object \(x\text{.}\)
Be careful not to inadvertently try to prove some property of members of the empty set! You will be proving a vacuously true statement. (See Section 4.3.)