Section 13.1 Basics
Objectives
Use sets and set notation to describe collections of elements.
An art collector might own a collection of paintings, while a music lover might keep a collection of CDs. Any collection of items can form a set.
Definition 13.1.1. Set.
A set is a collection of distinct objects, called elements of the set.
A set can be defined by describing the contents, or by listing the elements of the set, enclosed in curly brackets.
Example 13.1.2.
Some examples of sets defined by describing the contents:
a) The set of all even numbers
b) The set of all books written about travel to Chile
Some examples of sets defined by listing the elements of the set:
a) {1, 3, 9, 12}
b) {red, orange, yellow, green, blue, indigo, purple}
A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is equivalent to the set {3, 1, 2}.
Definition 13.1.3. Notation for Sets.
Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.
The symbol \(\in\) means “is an element of”.
A set that contains no elements, \(\{\; \}\text{,}\) is called the empty set and is notated \(\emptyset\text{.}\)
Example 13.1.4.
Let \(A = \{1, 2, 3, 4\}.\)
To notate that 2 is element of the set, we’d write \(2\in A\text{.}\)
Sometimes a collection might not contain all the elements of a set. For example, Chris owns three Madonna albums. While Chris’s collection is a set, we can also say it is a “subset” of the larger set of all Madonna albums.
Definition 13.1.5. Subset.
A subset of a set \(A\) is another set that contains only elements from the set \(A\text{,}\) but may not contain all the elements of \(A\text{.}\)
If \(B\) is a subset of \(A\text{,}\) we write \(B\subseteq A\text{.}\)
Definition 13.1.6. Proper Subset.
A proper subset is a subset that is not identical to the original set – it contains fewer elements.
If \(B\) is a proper subset of \(A\text{,}\) we write \(B\subset A\text{.}\)
Example 13.1.7.
Consider these three sets:
Here \(B\subset A\) since every element of \(B\) is also an even number, so is an element of \(A\text{.}\)
More formally, we could say \(B\subset A\) since if \(x\in B\text{,}\) then \(x\in A\text{.}\)
It is also true that \(B\subset C\text{.}\)
\(C\) is not a subset of \(A\text{,}\) since \(C\) contains an element, \(3\text{,}\) that is not contained in \(A\text{.}\)
Example 13.1.8.
Suppose a set contains the plays “Much Ado About Nothing”, “MacBeth”, and “A Midsummer Night's Dream”. What is a larger set this might be a subset of?
There are many possible answers here. One would be the set of plays by Shakespeare. This is also a subset of the set of all plays ever written. It is also a subset of all British literature.
Try It Now 13.1.9.
The set \(A = \{1, 3, 5\}\text{.}\) What is a larger set this might be a subset of?
There are several answers: The set of all odd numbers less than \(10\text{.}\) The set of all odd numbers. The set of all integers. The set of all real numbers.
Exercises 13.1.1 Exercises
1.
List out the elements of the set “The letters of the word Mississipi”.
\(\{m,i,s,p\}.\)
2.
List out the elements of the set “Months of the year”.
3.
Write a verbal description of the set \(\{3, 6, 9\}\text{.}\)
One possibility is: Multiples of 3 between 1 and 10.
4.
Write a verbal description of the set \(\{a, i, e, o, u\}\text{.}\)
5.
Is \(\{1, 3, 5\}\) a subset of the set of odd integers?
Yes.
6.
Is \(\{A, B, C\}\) a subset of the set of letters of the alphabet?
For problems 7-12, consider the sets below, and indicate if each statement is true or false.
7.
\(3\in B\text{.}\)
True.
8.
\(5\in C\text{.}\)
9.
\(B\subset A\text{.}\)
True.
10.
\(C\subset A\text{.}\)
11.
\(C\subset B\text{.}\)
False.
12.
\(C\subset U\text{.}\)