24 Basics
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.
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 1
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}.
Notation
Commonly, we will use a variable to represent a set, to make it easier to refer to that set later.
The symbol ∊ means “is an element of”.
A set that contains no elements, { }, is called the empty set and is notated ∅
Example 2
Let A = {1, 2, 3, 4}
To notate that 2 is element of the set, we’d write 2 ∊ A
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.
Subset
A subset of a set A is another set that contains only elements from the set A, but may not contain all the elements of A.
If B is a subset of A, we write B ⊆ A
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, we write B ⊂ A
Example 3
Consider these three sets
A = the set of all even numbers B = {2, 4, 6} C = {2, 3, 4, 6}
Here B ⊂ A since every element of B is also an even number, so is an element of A.
More formally, we could say B ⊂ A since if x ∊ B, then x ∊ A.
It is also true that B ⊂ C.
C is not a subset of A, since C contains an element, 3, that is not contained in A
Example 4
Suppose a set contains the plays Much Ado About Nothing, MacBeth, and A Midsummer’s Night 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 1
The set A = {1, 3, 5}. What is a larger set this might be a subset of?