2.1: Language of Sets
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 .
A set is a collection of distinct objects, called elements or members of the set
A set can be defined by listing the elements of the set, or by describing the contents in set-builder notation, enclosed in curly brackets.
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}
Some examples of sets defined by describing the contents in set-builder notation :
- C = {x : x is an even number}
- D = {y : y is a book written about travel to Chile}
A set simply specifies the contents; order is not important. The set represented by {1, 2, 3} is the same as the set {3, 1, 2}.
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, \(\{ \}\), is called the empty set or null set and is notated \(\emptyset\)
Let \(A=\{1,2,3,4\}\)
To notate that 2 is element of the set, we'd write \(2 \in A\)
A set is well-defined if we are able to tell whether any particular object is an element of the set.
Which sets are well-defined and which sets are not well-defined?
- A = {b : b is a type of tree}
- B = {g : g is a tasty food}
- C = {z : z is a restaurant in San Francisco}
Solution
The sets A and C are well-defined because we know exactly what types of trees there are and restaurants in San Francisco. The set B is not well-defined because there are different ideas of what a tasty food is.
The universal set is the set of all elements under consideration in a given discussion denoted by the letter U.
Example: U = {k : k is a student at Las Positas College}
Often times we are interested in the number of items in a set. This is called the cardinality of the set.
The number of elements in a set is the cardinal number of that set.
The cardinal number of the set \(A\) is often notated as \(n(A)\)
What is the cardinal number of \(\emptyset\)?
Solution
Since this is the empty set, \(n(\emptyset\))=0
What is the cardinal number of P = {h : h is the English name for the months of the year}?
Solution
The cardinal number, \(n(P)\)=12, since there are 12 months in the year