sets - university of british columbia department of ...elyse/220/2016/1sets.pdf · 1. sets 1.1...

1. Sets

1.1 Intro-duction toSets

1.2 TheCartesianProduct

1.3 Subsets

1.4 PowerSets

1.5 Union,Intersec-tion,Difference

1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Sets

First chapter: Sets

Slow start–introduction to sets, easy problems

Lead to more in-depth problems.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition

A set is a collection of things. The things in the set are called elements ofthe set.

Convention

We often write the elements of a set separated by commas inside curly braces.

Example:

{polar, brown, N.Am. black, Asiatic black, Andean, panda, sun, sloth}

Set: extant species of bear.

This is a finite set.It has a finite number of elements.

Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}

Ellipsis (. . .) means “continue the pattern.”

This is an infinite set. It has an infinite number of elements.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


Convention


Example:


Set: extant species of bear. This is a finite set.It has a finite number of elements.

Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}

Ellipsis (. . .) means “continue the pattern.”

This is an infinite set. It has an infinite number of elements.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


Convention


Example:


Set: extant species of bear. This is a finite set.It has a finite number of elements.

Example:P = {2, 3, 5, 7, 11, 13, 17, 19, . . .}

Ellipsis (. . .) means “continue the pattern.”This is an infinite set. It has an infinite number of elements.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition

Two sets are equal if they contain exactly the same elements.

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}

“element of” symbol

We write 10 ∈ A to mean “10 is an element of A”.

We write 10 6∈ A to mean “10 is not an element of A”.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}




How can you finish these?

0 ∈

B

7

6∈

A

35, 45, 55 ∈

E This is a common “abuse of notation”

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}





0 ∈ B

7

6∈

A

35, 45, 55 ∈


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}





0 ∈ B

7 6∈ A

35, 45, 55 ∈


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}





0 ∈ B

7 6∈ A

35, 45, 55 ∈ E

This is a common “abuse of notation”

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}





0 ∈ B

7 6∈ A

35, 45, 55 ∈ E This is a common “abuse of notation”

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}




We write |A| to mean the size, or cardinality of a set1 A. For finite sets,

this means the number of distinct elements.

1For now, we’ll only talk about the cardinalities of finite sets

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}







|A| =

4

|A| =∣∣∣

D

∣∣∣1For now, we’ll only talk about the cardinalities of finite sets

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}







|A| = 4

|A| =∣∣∣

D

∣∣∣1For now, we’ll only talk about the cardinalities of finite sets

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}







|A| = 4

|A| =∣∣∣ D ∣∣∣


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}






True or false: C=3

False. C is a set, 3 is a number.

True or false: If two sets are equal, their cardinalities are equal.

True.


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}






True or false: C=3 False. C is a set, 3 is a number.


True.


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definitions

Definition


A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

D = {20, 15, 10, 5} E = {5, 10, 15, 20, . . .}






True or false: C=3 False. C is a set, 3 is a number.


True.1For now, we’ll only talk about the cardinalities of finite sets

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.

Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”

the Natural Numbers, N .

N = {1, 2, 3, 4, 5, . . .}

This is an infinite set.∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)

the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}

No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.

∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}

|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| =

0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0

True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅.

True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.

True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.

False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.

“An empty box is not the same as a box with an empty box inside it.”


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets

the Empty Set, ∅ . This is the set with no elements.Often, in a proof, you will say a theorem holds for every nonempty set.∅ = {}|∅| = 0True or false: If |A| = 0, then A = ∅. True.True or false: ∅ = {∅}.False. The set {∅} has one element: the empty set.“An empty box is not the same as a box with an empty box inside it.”

∅ {∅}


N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}

This is an infinite set.

∞ 6∈ NControversy: is 0 a natural number? (In our text, it is not.)

the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}

This is an infinite set.∞ 6∈ N

Controversy: is 0 a natural number? (In our text, it is not.)

the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}

No controversy: 0 is definitely an integer

Z stands for “number” if you’re German (Zahl)Which is bigger, N or Z? stay tuned

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}

No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)

Which is bigger, N or Z? stay tuned

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}

No controversy: 0 is definitely an integerZ stands for “number” if you’re German (Zahl)Which is bigger, N or Z?

stay tuned

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Special Sets



N = {1, 2, 3, 4, 5, . . .}


the Integers, Z .

Z = {. . . ,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, . . .}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Set-Builder Notation

A︸︷︷︸name

= { expression︸︷︷︸all things of the form

: rule︸︷︷︸such that

}

Examples:

{n : n is prime} (set of all primes)

E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers

{n ∈ N :√n ∈ N} set of all perfect squares

Set of all odd numbers:

{2n + 1 : n ∈ Z}

All integers not divisible by 3:

{n ∈ Z :

n

36∈ Z

}

True or false: the following sets are equal.

{ −2n : n ∈ Z and −2n > 7 }{ 2(n + 1) : n ∈ N and n > 2 }{ 2n : n ∈ Z and n ≥ 4 }

True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:

{n : n is prime}

(set of all primes)




{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:


E = {2n : n ∈ N}

= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}Positive even numbers



{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:


E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}

= {2, 4, 6, 8, . . .}Positive even numbers



{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:


E = {2n : n ∈ N}= {2(1), 2(2), 2(3), 2(4), . . .}= {2, 4, 6, 8, . . .}

Positive even numbers



{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:





{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:



{n ∈ N :√n ∈ N}

set of all perfect squares


{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:





{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:





{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}True or false: the following sets are equal.


True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:





{2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:




Set of all odd numbers: {2n + 1 : n ∈ Z}


{n ∈ Z :

n

36∈ Z

}



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:





All integers not divisible by 3:{n ∈ Z :

n

36∈ Z



True: {8, 10, 12, 14, 16, 18, . . .}

{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:






n

36∈ Z



True: {8, 10, 12, 14, 16, 18, . . .}{a− b : a, b ∈ N} =

Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


A︸︷︷︸name



}

Examples:






n

36∈ Z



True: {8, 10, 12, 14, 16, 18, . . .}{a− b : a, b ∈ N} = Z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}

Rational Numbers (ratios of integers)

R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =

{x ∈ R : a < x < b}

open interval[a, b) =

{x ∈ R : a ≤ x < b}

half-closed interval[a,∞) =

{x ∈ R : x ≥ a}

infinite interval

Exercises: On the xy -plane, sketch the points in the sets below.

{(x , y) : x = y}{(x , y) : |x | < 1, |y | < 2}{(2x , y) : x < y}{(x , y) : x , y ∈ N}{(x , x + y) : x ∈ R, y ∈ Z}

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)

[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =

{x ∈ R : a < x < b}


{x ∈ R : a ≤ x < b}


{x ∈ R : x ≥ a}

infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b}

closed interval(a, b) =

{x ∈ R : a < x < b}


{x ∈ R : a ≤ x < b}


{x ∈ R : x ≥ a}

infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval

(a, b) =

{x ∈ R : a < x < b}


{x ∈ R : a ≤ x < b}


{x ∈ R : x ≥ a}

infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) =

{x ∈ R : a < x < b}


{x ∈ R : a ≤ x < b}


{x ∈ R : x ≥ a}

infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) =

{x ∈ R : a ≤ x < b}


{x ∈ R : x ≥ a}

infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) ={x ∈ R : a ≤ x < b} half-closed interval[a,∞) =

{x ∈ R : x ≥ a}

infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

More Special Sets

Q ={mn

: m, n ∈ Z and n 6= 0}


R: Real numbers (set of all numbers on the number line)[a, b] = {x ∈ R : a ≤ x ≤ b} closed interval(a, b) ={x ∈ R : a < x < b} open interval[a, b) ={x ∈ R : a ≤ x < b} half-closed interval[a,∞) ={x ∈ R : x ≥ a} infinite interval



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Motivation and Definition

Suppose we have sets L and T , where L is a set of locations and T is a set oftimes. We want to pair them, so that we have a new set of elements that tellsus both time and location.

Definition

An ordered pair is a list (x , y) of two things (x and y) such that

(x , y) = (a, b) if and only if x = a AND y = b. That is: order matters.

You have seen these used to describe points in the xy -plane.

Definition

The Cartesian product of two sets A and B is another set, denoted as

A× B (“A cross B”) and defined as

A× B = {(a, b) : a ∈ A and b ∈ B}.

So, T × L is a set, whose elements have the form (t, l):ordered pairs of times and locations.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Cartesian Product

We can visualize the pairs as a grid, similar to the xy -plane.Suppose T = {noon, 4pm, 8pm} andL = {in a box, with a fox, in a house, with a mouse}.

no

on

4p

m

8p

m

box

fox

house

mouse

(4pm,box) (8pm,box)(noon,box)

(4pm,fox) (8pm,fox)(noon,fox)

(4pm,house) (8pm,house)(noon,house)

(4pm,mouse) (8pm,mouse)(noon,mouse)

T

L

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Cartesian Products

Theorem

Suppose A and B are finite sets. Then

|A× B| =

|A||B|

Sketch the following sets:

R× {−1, 1}

x

y

Z× N

x

y

Z× R

x

y

The xy -plane is sometimes called R2, since it is R× R.You might have also seen xyz-coordinates called R3.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Cartesian Products

Theorem


|A× B| = |A||B|


R× {−1, 1}

x

y

Z× N

x

y

Z× R

x

y


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Cartesian Products

Theorem


|A× B| = |A||B|


R× {−1, 1}

x

y

Z× N

x

y

Z× R

x

y

The xy -plane is sometimes called R2, since it is R× R.

You might have also seen xyz-coordinates called R3.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Cartesian Products

Theorem


|A× B| = |A||B|


R× {−1, 1}

x

y

Z× N

x

y

Z× R

x

y


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Definition

An ordered triple is a list of three things, such as (x , y , z).

(a, b, c)

We take the Cartesian product of n sets A1, A2, . . . , An in exactly the wayyou’d imagine:

A1 × A2 × · · · × An = {(a1, a2, . . . , an) : ai ∈ Ai for all i = 1, 2, . . . , n}

We write An = A× A× · · · × A︸︷︷︸n times

Example

The dimensions of a cardboard box can be written as (x , y , z) ∈ R3.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C =

{(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.

Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|?

(3)(2)(2)(3) = 36

Theorem

|A1 × A2 × · · · × An| =

|A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.


(3)(2)(2)(3) = 36

Theorem

|A1 × A2 × · · · × An| =

|A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.


(3)(2)(2)(3) = 36

{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}

(1, (2, (3,

(1,x (1,y (2,x (2,y (3,x (3,y

(1,x,A, (1,x,B, (1,y,A, (1,y,B, (2,x,A, (2,x,B, (2,y,A, (2,y,B, (3,x,A, (3,x,B, (3,y,A, (3,y,B,

(1,x,A,30)(1,x,A,20)(1,x,A,10)

(1,x,B,30)(1,x,B,20)(1,x,B,10)

(1,y,A,30)(1,y,A,20)(1,y,A,10)

(1,y,B,30)(1,y,B,20)(1,y,B,10)

(2,x,A,30)(2,x,A,20)(2,x,A,10)

(2,x,B,30)(2,x,B,20)(2,x,B,10)

(2,y,A,30)(2,y,A,20)(2,y,A,10)

(2,y,B,30)(2,y,B,20)(2,y,B,10)

(3,x,A,30)(3,x,A,20)(3,x,A,10)

(3,x,B,30)(3,x,B,20)(3,x,B,10)

(3,y,A,30)(3,y,A,20)(3,y,A,10)

(3,y,B,30)(3,y,B,20)(3,y,B,10)

Theorem

|A1 × A2 × · · · × An| =

|A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.


(3)(2)(2)(3) = 36

{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,

(1,x (1,y (2,x (2,y (3,x (3,y


(1,x,A,30)(1,x,A,20)(1,x,A,10)

(1,x,B,30)(1,x,B,20)(1,x,B,10)

(1,y,A,30)(1,y,A,20)(1,y,A,10)

(1,y,B,30)(1,y,B,20)(1,y,B,10)

(2,x,A,30)(2,x,A,20)(2,x,A,10)

(2,x,B,30)(2,x,B,20)(2,x,B,10)

(2,y,A,30)(2,y,A,20)(2,y,A,10)

(2,y,B,30)(2,y,B,20)(2,y,B,10)

(3,x,A,30)(3,x,A,20)(3,x,A,10)

(3,x,B,30)(3,x,B,20)(3,x,B,10)

(3,y,A,30)(3,y,A,20)(3,y,A,10)

(3,y,B,30)(3,y,B,20)(3,y,B,10)

Theorem

|A1 × A2 × · · · × An| =

|A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.

Another example:Let A = {1, 2, 3}, B = {x , y}, C = {A,B}, and D = {10, 20, 30}.What is |A× B × C × D|? (3)(2)(2)(3) = 36

{(a, b, c, d) : a ∈ A, b ∈ B, c ∈ C , d ∈ D}(1, (2, (3,

(1,x (1,y (2,x (2,y (3,x (3,y


(1,x,A,30)(1,x,A,20)(1,x,A,10)

(1,x,B,30)(1,x,B,20)(1,x,B,10)

(1,y,A,30)(1,y,A,20)(1,y,A,10)

(1,y,B,30)(1,y,B,20)(1,y,B,10)

(2,x,A,30)(2,x,A,20)(2,x,A,10)

(2,x,B,30)(2,x,B,20)(2,x,B,10)

(2,y,A,30)(2,y,A,20)(2,y,A,10)

(2,y,B,30)(2,y,B,20)(2,y,B,10)

(3,x,A,30)(3,x,A,20)(3,x,A,10)

(3,x,B,30)(3,x,B,20)(3,x,B,10)

(3,y,A,30)(3,y,A,20)(3,y,A,10)

(3,y,B,30)(3,y,B,20)(3,y,B,10)

Theorem

|A1 × A2 × · · · × An| =

|A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.


Theorem

|A1 × A2 × · · · × An| =

|A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

Example

Let A = {1, 2}, B = {3}, and C = {1, 3}. Then

A× B × C = {(1, 3, 1), (1, 3, 3), (2, 3, 1), (2, 3, 3)}

Note |A× B × C | = 4.


Theorem

|A1 × A2 × · · · × An| = |A1||A2| · · · |An|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

What shape does [−1, 1]3 make?

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Higher Orders

What shape does [−1, 1]3 make? [−1, 1]3 = [−1, 1]︸︷︷︸x

× [−1, 1]︸︷︷︸y

× [−1, 1]︸︷︷︸z

x

y

z

x

y

z

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Definition and First Examples

Definition

A set A is a subset of a set B is every element of A is also an element of B.We write

A ⊆ B

If A is not a subset of B, we write

A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

A ⊆ B C ⊆ A,B True or false: A ⊆ A True

True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.

False (see next question)

If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ?

X = Y (and so |X | = |Y |)

True or false: for any set X , ∅ ⊆ X .

True

If X ⊆ Y and Y ⊆ X , what do you know about X and Y ?

X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

A ⊆

B C ⊆ A,B True or false: A ⊆ A True




X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

A ⊆ B

C ⊆ A,B True or false: A ⊆ A True




X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

A ⊆ B C ⊆

A,B True or false: A ⊆ A True




X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

A ⊆ B C ⊆ A,B

True or false: A ⊆ A True




X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}

A ⊆ B C ⊆ A,B True or false: A ⊆ A

True




X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}





X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}





X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X .

True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}





X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}


True or false: if X and Y are finite sets, and X ⊆ Y , then |X | < |Y |.False (see next question)


X = Y (and so |X | = |Y |)


True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}



If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ? X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X .

True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}



If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ? X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X . True


X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


Definition


A ⊆ B


A 6⊆ B

A = {5, 10, 15, 20} B = {0, 5, 10, 15, 20} C = {5, 10, 15}



If X and Y are finite sets, X ⊆ Y and |X | ≥ |Y |, what do you know aboutthe relationship between X and Y ? X = Y (and so |X | = |Y |)True or false: for any set X , ∅ ⊆ X . True

If X ⊆ Y and Y ⊆ X , what do you know about X and Y ? X = Y (!!!)

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Notation Review

True or False?

1 ⊆ {1, 2, 3}

False

1 ∈ {1, 2, 3}

True

{1} ⊆ {1, 2, 3}

True

{1} ∈ {1, 2, 3}

False

1 ∈ {{1}, {2, 3}}

False

{1} ∈ {{1}, {2, 3}}

True

1 ⊆ {{1}, {2, 3}}

False

{1} ⊆ {{1}, {2, 3}}

False

{{1}} ⊆ {{1}, {2, 3}}

True

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Notation Review

True or False?

1 ⊆ {1, 2, 3} False

1 ∈ {1, 2, 3} True

{1} ⊆ {1, 2, 3} True

{1} ∈ {1, 2, 3} False

1 ∈ {{1}, {2, 3}} False

{1} ∈ {{1}, {2, 3}} True

1 ⊆ {{1}, {2, 3}} False

{1} ⊆ {{1}, {2, 3}} False

{{1}} ⊆ {{1}, {2, 3}} True

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets

Example: List all subsets of the set S = {1, 2, 3}.

size sets notes0

∅ subset of any set

1

{1} {2} {3}

2

{2, 3} {1, 3} {1, 2} delete one element

3

{1, 2, 3}

Example: List all subsets of the set T = {a, b, {1, 2}}.Note: |T | = 3.

size sets notes0


1

{a} {b} {{1, 2}}

2

{b, {1, 2}} {a, {1, 2}} {a, b} delete one element

3

{a, b, {1, 2}}

How many subsets does any set of cardinality 3 have? 8

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1

{1} {2} {3}

2


3

{1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1 {1} {2} {3}2


3

{1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1 {1} {2} {3}2

{2, 3} {1, 3} {1, 2}

delete one element3

{1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3}

{1, 3} {1, 2}

delete one element3

{1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3}

{1, 2}

delete one element3

{1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3

{1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets


size sets notes0 ∅ subset of any set1 {1} {2} {3}2 {2, 3} {1, 3} {1, 2} delete one element3 {1, 2, 3}


size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets



Example: List all subsets of the set T = {a, b, {1, 2}}.

Note: |T | = 3.

size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets




size sets notes0


1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets




size sets notes0 ∅ subset of any set1

{a} {b} {{1, 2}}

2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets




size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2


3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets




size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2 {b, {1, 2}} {a, {1, 2}} {a, b} delete one element3

{a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets




size sets notes0 ∅ subset of any set1 {a} {b} {{1, 2}}2 {b, {1, 2}} {a, {1, 2}} {a, b} delete one element3 {a, b, {1, 2}}


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets





How many subsets does any set of cardinality 3 have?

8

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Listing Subsets






1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets

Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?

We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .

S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }

Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).

There are 2n subsets of a set of size n.

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets

Let n be a natural number, or zero. How many subsets does a set ofcardinality n have?We already saw how to count ordered n-tuples, (a1, a2, . . . , an) with ai ∈ Ai .

S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }

T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }

T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )

U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }

Every subset of our set of size n corresponds to some unique ordered n-tuple(with entries from {0, 1}), and every n-tuple corresponds to some uniquesubset.

The number of subsets of a set of size n is the same as the size of {0, 1}n(the number of ordered n-tuples where every entry is 0 or 1).


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets


S = { 1 , 2 , 3 , 4 , 5 }T = { 2 , 4 , 5 }T = ( 0 , 1 , 0 , 1 , 1 )

U = ( 0 , 0 , 1 , 1 , 0 )U = { 3 , 4 }



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Number of Subsets – Check


n set subsets 2n

0

∅ ∅

20 = 1

1

{1} ∅, {1}

21 = 2

2

{1, 2} ∅, {1}, {2}, {1, 2}

22 = 4

3

{1, 2, 3} (already done: 8 of them)

23 = 8

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0

∅ ∅

20 = 1

1

{1} ∅, {1}

21 = 2

2

{1, 2} ∅, {1}, {2}, {1, 2}

22 = 4

3 {1, 2, 3}

(already done: 8 of them)

23 = 8

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0

∅ ∅

20 = 1

1

{1} ∅, {1}

21 = 2

2

{1, 2} ∅, {1}, {2}, {1, 2}

22 = 4

3 {1, 2, 3} (already done: 8 of them) 23 = 8

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0

∅ ∅

20 = 1

1 {1}

∅, {1}

21 = 2

2

{1, 2} ∅, {1}, {2}, {1, 2}

22 = 4


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0

∅ ∅

20 = 1

1 {1} ∅, {1} 21 = 2

2

{1, 2} ∅, {1}, {2}, {1, 2}

22 = 4


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0

∅ ∅

20 = 1

1 {1} ∅, {1} 21 = 2

2 {1, 2}

∅, {1}, {2}, {1, 2}

22 = 4


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0

∅ ∅

20 = 1

1 {1} ∅, {1} 21 = 2

2 {1, 2} ∅, {1}, {2}, {1, 2} 22 = 4


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0 ∅

∅

20 = 1

1 {1} ∅, {1} 21 = 2

2 {1, 2} ∅, {1}, {2}, {1, 2} 22 = 4


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets



n set subsets 2n

0 ∅ ∅ 20 = 1

1 {1} ∅, {1} 21 = 2

2 {1, 2} ∅, {1}, {2}, {1, 2} 22 = 4


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition

If A is a set, the power set of A is the set of all subsets of A.

We write P(A) for the power set of A,

P(A) = {X : X ⊆ A}

Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}

True or False: For every set A, A ⊆ P(A)True or False: For every set A, A ∈ P(A)

Theorem

For any finite set A,|P(A)| = 2|A|

There are 193 member nations in the UN. How many ways are there to form acommittee made up of at least two member nations?

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}

Example: P ({a, b, c}) =

{∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},{a, b, c}}


Theorem



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}

Example: P ({a, b, c}) ={∅,

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


Theorem



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}

Example: P ({a, b, c}) ={∅,{a}, {b}, {c},

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


Theorem



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}

Example: P ({a, b, c}) ={∅,{a}, {b}, {c},{b, c}, {a, c}, {a, b},

{a, b, c}}


Theorem



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}



Theorem



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}



Theorem

For any finite set A,|P(A)| =

2|A|


1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Power Sets

Definition



P(A) = {X : X ⊆ A}



Theorem



1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

Fun Context

Number of subcommittees of the UN consisting of at least two membernations:

2193︸︷︷︸total number of subsets

− 1︸︷︷︸empty set

− 193︸︷︷︸single-element subsets

≈ 2193

population of humans ≈ 7 billion ≈ 232.7

Number of atoms in the earth2 ≈ 1050 ≈ 2166

Number of stars in the observable universe3 ≈ 1021 ≈ 270

Number of atoms in the Milky Way4 ≈ 1068 ≈ 2256

Number of atoms in the known universe5 ≈ 1080 ≈ 2266

How many different subcommittees can Canada be on?

2http://education.jlab.org/qa/mathatom_05.html,http://www.fnal.gov/pub/science/inquiring/questions/atoms.html

3http://scienceline.ucsb.edu/getkey.php?key=37754https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy5http://www.universetoday.com/36302/atoms-in-the-universe/

http://education.jlab.org/qa/mathatom_05.html

http://www.fnal.gov/pub/science/inquiring/questions/atoms.html

http://scienceline.ucsb.edu/getkey.php?key=3775

https://www.quora.com/How-many-atoms-are-there-in-the-milky-way-galaxy

http://www.universetoday.com/36302/atoms-in-the-universe/

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

True or False: If A is a finite set, then P(A) is finite.

True (even though it can be huuuuuge)

True or False: For every finite set A, |A| ≤ |P(A)|.

True. (Even when A = ∅.)

If A and B are finite sets, calculate the following cardinalities:

|P(A)× B|

= 2|A| · |B|

|P(A× B)|

= 2|A||B|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

True or False: If A is a finite set, then P(A) is finite.True (even though it can be huuuuuge)

True or False: For every finite set A, |A| ≤ |P(A)|.

True. (Even when A = ∅.)


|P(A)× B|

= 2|A| · |B|

|P(A× B)|

= 2|A||B|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets


True or False: For every finite set A, |A| ≤ |P(A)|.True. (Even when A = ∅.)


|P(A)× B|

= 2|A| · |B|

|P(A× B)|

= 2|A||B|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets




|P(A)× B| = 2|A| · |B|

|P(A× B)|

= 2|A||B|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets




|P(A)× B| = 2|A| · |B|

|P(A× B)| = 2|A||B|

1. Sets



1.3 Subsets

1.4 PowerSets


1.6 Com-plement

1.7 VennDiagrams

1.8 IndexedSets

For Sections 1.5-1.8, see the next set of slides.

http://www.math.ubc.ca/~elyse/220/2016/2Sets.pdf

sets - university of british columbia department of ...elyse/220/2016/1sets.pdf · 1. sets 1.1...

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