Course: Math LiteracyAim: Set Notation

Aim: How do we deal with chaos?

Do Now:

How many of you drive yourself to school?

How many of you take the bus to school?

How many of you walk to school?

How many of you are driven to school?

How many of you get to school by other means?

Course: Math LiteracyAim: Set Notation

Terminology of sets

Sets deal with sorting of objects into similar groupings, allowing us to order and structure the world. Our minds cannot find order and meaning without creating collections.

•Objects in sets are called members or elements.

•A set is a collection of objects whose contents can be clearly determined.

ex. set of days of week whose elements include Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday.

Course: Math LiteracyAim: Set Notation

Terminology of sets

The set of the days of the week are the elements Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday.

Three methods commonly used to designate a set.

word description

W = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

roster method

order not important

only brackets and commas

Course: Math LiteracyAim: Set Notation

Terminology of sets

Three methods commonly used to designate a set.

set builder notation

“Set W is the set of all elements x such that x is a day of the week.”

W = {x | x is a day of the week}

Set of W

is all elements

x

the set of

such that

Course: Math LiteracyAim: Set Notation

Model Problem

Express set A = {x | x is a month that begins with the letter M} using the roster method.

A = {March, May}

Express set A = {x | x is a month that begins with the letter R} using the roster method.

A = { } or

Empty Set – also called the null set, is the set that contains no elements.

Note: {} is a set containing the element

Course: Math LiteracyAim: Set Notation

Terminology of sets

well defined – a set must be well defined, meaning the contents are clearly determined

ex. the collection of actors who have won Academy Awards is a set.

ex. the collection of actors great actors.

well defined

not well defined

subject to interpretation

Course: Math LiteracyAim: Set Notation

Terminology of sets

The symbol is used to indicate that an object is an element of a set. It is used to replace the words, “is an element of.”

The symbol is used to indicate that an object is not an element of a set. It is used to replace the words, “is not an element of.”

r {a, b, c, . . . , z}

ellipsiselements of set continue in same manner up to and including the z.

7 {1, 2, 3, 4, 5}

Course: Math LiteracyAim: Set Notation

Model Problem

True or False?

8 {1, 2, 3, . . . , 10}

r {a, b, c, z}

{Monday} {x | x is a day of the week}

T

T

T

Course: Math LiteracyAim: Set Notation

Terminology of sets

N = {1, 2, 3, 4, 5, . . .} is the set of Natural Numbers.

Express the following sets using the roster method.

Set A is the set of natural numbers less than 5.

A = {1, 2, 3, 4}B is the set of natural numbers greater than or equal to 25.

A = {25, 26, 27, 28, . . .}

E = {x|x N and x is even}E = {2, 4, 6, 8, . . .}

F = {x|x N and x is between 6 and 10}F = {7, 8, 9}

Course: Math LiteracyAim: Set Notation

Terminology of sets

Cardinal Number – the number of elements in a set. Also called the cardinality of the set.

The cardinal number of set, represented by n(A), is the number of distinct elements is set A. The

symbol n(A) is read “n of A.”

ex. Z = {a, e, i, o, u}set Z has a cardinality of 5 or n(Z) = 5

Find the cardinal number of the following sets.

A = {7, 9, 11, 13} B = {0}

C = {13, 14, 15, . . . , 22, 23}

n(A) = 4 n(B) = 1

n(C) = 11

n() = 0

Course: Math LiteracyAim: Set Notation

Terminology of sets

Finite Set – if n(A) = 0 or n(A) is a natural number. A set that is not finite is called an infinite set.

{3, 6, 9, 12, . . . } is an infinite set and has no cardinality

Equal Sets – contain exactly the same elements, regardless of order or possible repetition of

elements. Equality of sets is symbolized as A = B

A = {w, x, y, z} and B = {z, y, x, w}

A = B

n(A) = n(B)

Course: Math LiteracyAim: Set Notation

Terminology of sets

Equivalent Sets – contain the same number of elements. Cardinalities are equal: n(A) = n(B)

A = {The Graduate, The Godfather, Titanic} B = {Hoffman, Brando, DiCaprio}

Equal, equivalent, or both?

Not equal – do not contain exactly the same elements

Equivalent – each contains three elements: n(A) = n(B)

note: repeating elements in a set do not add new elements to the set.

A = {1, 1, 2, 2, 3} and B = {1, 2, 3}; A = B; n(A) = n(B)

Course: Math LiteracyAim: Set Notation

Terminology of sets

Subsets – A is a subset of set B, expressed as A B if every element of set A is also an element of

set B. means ‘is not a subset’

Let A = {x|x is a planet in the Earth’s solar system.

B = {Mercury, Venus, Earth, Mars, Jupiter}

C = {1 Ceres}

B A C A

1 Ceres is the solar system’s largest asteroid. It is 974 km in diameter and contains about 25% of the

mass of all the asteroids combined.

Course: Math LiteracyAim: Set Notation

Study Tip

Relationships between 2 Sets

Proper Subset – Set A is a proper subset of set B, expressed as A B, if set A is a subset of set B and sets A and B are not equal (A B)

A B

A is a subset of B

a < ba is less than or

equal to b

A BA is a proper

subset of B

a < b

a is less than b but not equal to b

Course: Math LiteracyAim: Set Notation

Model Problems

Because every twentieth century US president was male, this means that the set of twentieth century US presidents is a subset of males.

{x | x is a 20th century US President} {x | x is a male

Write or in each blank

A = {1, 3, 5, 7} B = {1, 3, 5, 7, 9, 11} A ____ B

A = {set of music written in 20th century} B = {set of jazz music} A ____ B

Course: Math LiteracyAim: Set Notation

Number of Subsets

1. For any set B, B.2. For any set B other than the empty set, B.

SetNumber of Elements

List of All SubsetsNumber of

Subsets

{ } 0 { } 1

{ a } 1 { } { a } 2

{ a, b } 2 { } { a } { b } { a, b } 4

{ a, b, c } 3{ } { a } { b } { c } { a, b } { b, c } { a, c }

{ a, b, c }8

The number of subsets of a set with n elements is 2n.

The number of proper subsets of a set is 2n – 1.

Course: Math LiteracyAim: Set Notation

Model Problem

You recently purchase three books: {The Color Purple, Hannibal, The Royals}

You are deciding which books, if any to take on vacation. You have enough room to pack up to three books, but may take fewer or not at all.

a. Find the number of subsets of the given set.

b. List all the subsets.

c. How many of the subsets are proper subsets?

n = 3; 23 = 8

The number of subsets of a set with n elements is 2n.

The number of proper subsets of a set is 2n – 1.

23 – 1 = 7

Course: Math LiteracyAim: Set Notation