2.1: Language of Sets

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May 26, 2022
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- 2: Set Theory
- 2.2: Comparing Sets

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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 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.

Example 1

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}.

Set Notation

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 $\emptyset$

Example 2

Let $A=\{1,2,3,4\}$

To notate that 2 is element of the set, we'd write $2 \in A$

Well-defined

A set is well-defined if we are able to tell whether any particular object is an element of the set.

Example 3

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.

Universal Set

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.

Cardinal Number

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 $n(A)$

Example 4

What is the cardinal number of $\emptyset$ ?

Solution

Since this is the empty set, $n(\emptyset$ )=0

Example 5

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

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Set

Example 1

Set Notation

Example 2

Well-defined

Example 3

Universal Set

Cardinal Number

Example 4

Example 5

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