02 LCD Slide Handout 1(6)

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Set Theory

Null, Singleton, and Universal Set

What is a Set?

Writing Sets

Relationship among Sets

Notes on the Concept of Sets

Operation on Sets

G0001

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What is a Set ?Set

*Property of STI G0001

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What is a Set ?Set is a collection of

distinct objects.


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is a collection of

distinct objects.Set September, October,

November, December


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Element is a particular

member of a set.Cardinality is the number of

elements in a set.*Property of STI G0001

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Writing Sets


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Two ways of writing sets:

1. Roster Method

2. Set Builder Notation


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Example

{September, October, November, December}

{Monday, Tuesday, Wednesday, Thursday,

Friday, Saturday, Sunday}

{Pres. Ramos, Pres. Estrada, Pres. Arroyo

Pres. Aquino}

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

{. . . −6, −3, 0, 3, 6, . . . }


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Set Builder Notation


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Example

{September, October, November, December}

{x | x is a month ending in “ber”}

ROSTER METHOD:

SET BUILDER NOTATION:

the set of all x such that x

is a month ending in “ber”such that


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Example

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

{x | x is a positive integer}

ROSTER METHOD:


{x | x is an integer and x > 0}


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Example

{2, 4, 6, 8, 10, . . . }

{x | x is a positive even number}

ROSTER METHOD:


{x | x = 2k , where k is a positive integer}


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Membership


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A

a ∈ A

a

a is an element of A*Property of STI G0001

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A

m ∉ A

m

m is not an element of A*Property of STI G0001

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Example

{September, October, November, December}B =

September

August

April

November

∈ B

∈ B

∉ B

∉ B


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CheckPoint

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


C = {x | x is an integer and 1 ≤ x ≤ 5}

136 ∈ C ∈ C ∉ C 7∉ C *Property of STI G0001

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CheckPoint

ROSTER METHOD:

D = {. . .−

10,−

5, 0, 5, 10, . . .}

2510012 ∈ D ∈ D ∉ D 16 ∉ D


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Exercise

1. A = {. . . −3, −2, −1, 0, 1, 2, 3, . . . }

2. B = {x | x is an even number and 3 < x < 9}

5. E = {−2, −1, 0, 1, 2}

4. D = {x | x is an odd number}

3. C = {2, 3, 5, 7, 11, 13, . . .}


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Notes on the

Concept of Sets


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Sets are invariant under the

following:

1. ordering of elements

2. repetition of elements

3. properties defining the set


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ordering of elements{1, 2, 3} is the same as {2, 1, 3}

{a , b , c } is the same as {c , b , a }


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{1, 2, 3} is the same as {2, 1, 3}

{a , b , c } is the same as {c , b , a }

ordering of elements

{1, 1, 2} is the same as {1, 2}

{b , b , b } is the same as {b }


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repetition of elements

{1, 1, 2} is the same as {1, 2}

{b , b , b } is the same as {b }

{x | x is a positive even number}

is the same as

{x | x = 2k, where k is a positive integer}


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Null, Singleton, and

Universal Set


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Null Set is a set with

no element.

Notation: or { }


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Singleton

Set

is a set with

one element.

Example:

{a} {1}

{ x | x is an integer and 1 < x < 3}


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Universal

Set

is a set containing

all elements under

consideration

Notation: U


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Relationship

among Sets


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U

A BVenn Diagram=


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U

A Bdisjoint

A and B are said to be disjoint if A and B has no

common elements.

Let A and B be sets.

efinition

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

U = {x | x is a real number}

Example

B = {6, 7, 8, 9, 10}

A = {a , b , c } B = {e , f , g , h , k }

A = {x | x is a letter and x is a consonant}

B = {x | x is a letter and x is a vowel}*Property of STI G0001

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U

A Bintersecting

A and B are said to be intersecting if A and B have at

least one common element.


efinition

A = {2, 4, 6}

U = {x | x is a real number}

Example

B = {3, 6, 9, 12, 15}

A = {a, b, c }

A = {x | x is a prime number}

B = {b , c, d, e , g }

B = {x | x is an odd number}*Property of STI G0001

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U

A = Bequal

A and B are said to be equal if A and B have the same

elements.


efinition

A = {a, b, c }

A = {x | x is an even number }

Example

B = {b , c, a }

B = {x | x is divisible by 2}*Property of STI G0001

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UAproper subset

proper superset

Let A and B be sets. B is a proper subset of A (or A is a proper

superset of B ) if:

efinition

B

a. all elements of B is in A

b. not all elements of A is in B

B ⊂ A or A ⊃ B

Notation

A = {a, b, c }

A = {x | x is an integer }

Example

B = {a , b, c, d, e }

B = {x | x is an even number}*Property of STI G0001

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subset / superset

B is a subset of A

B ⊂ A

A is a superset of B

B = A


efinition

B is a subset of A (or A is a superset of B ) if the following

condition is satisfied:

All elements of B is in A.


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B ⊆ Aotation

subset / superset


efinition

A is equal to B if the following conditions are satisfied:

a. A ⊆ B

b

. B ⊆ A

A ⊇ B or


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CheckPoint

C = {2, 4, 6, 8 . . .} D = {3, 6, 9, 12, 15 . . .}

INTERSECTING


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CheckPoint

DISJOINT


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Exercise

1. A = {x | x is an integer}B = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}

3. E = {2, 3, 4, 5, 6, . . .}

D = {. . . −6, −3, 0, 3, 6, . . .}2. C = {. . . −5, −2, 1, 4, 7, 10, . . .}

B ⊂ A

D = {x | x is an integer and x > 1}

4. E = {x | x is a prime number}D = {x | x is an even number}

disjoint

equal

intersecting


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Operation on

Sets


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A

C

B

O


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Set Operations

1. Union

2. Intersection

3. Relative Complement

(A ∪ B )

(A B )

(A B )


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Example:

A = {1, 2, 3} B = {3, 4}

A ∪ B = {1, 2, 3, 4}

D = {a , b, c, d } G = {c , d , e , f , g }

D ∪ G = {a , b , c , d , e , f , g }

W = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}

W ∪ Q = {x | x is an integer}

Q = {. . . −5, −3, −1, 1, 3, 5, . . .}


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Intersection (A B )


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Example:

A = {1, 2, 3} B = {3, 4}

A B = {3}

D = {a , b, c, d } G = {c , d , e , f , g }

D G = {c , d }

W = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}

W ∪

Q = { } =

Q = {. . . −5, −3, −1, 1, 3, 5, . . .}


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Relative Complement (A B )


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Example:

A = {1, 2, 3} B = {3, 4}

A B = {1, 2}

D = {a , b, c, d } G = {c , d , e , f , g }

D G = {a , b }

W = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}

W Q = W

Q = {. . . −5, −3, −1, 1, 3, 5, . . .}


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Complement


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Example:

U = {1, 2, 3, 4, 5} A = {3, 4}

A ‘ = {1, 2, 5}

D = {x | x is consonant }

D ‘ = {a , e, i, o, u }

U = {x | x is an integer}

Q ‘ = {. . . −6, −4, −2, 0, 2, 4, 6, . . .}

Q = {. . . −5, −3, −1, 1, 3, 5, . . .}

U = {x | x is a letter in English alphabet}


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CheckPoint

C = {2, 4, 6, 8 . . .} D = {1, 3, 5, 7, 9 . . .}

C D = {x | x is an integer and x > 1}

C D = { }C D = C D C = D

U = {x | x is an integer}

C ‘ = {. . . −3, −2, −1, 0} D

D ‘ = {. . . −3, −2, −1, 0} C *Property of STI G0001

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Exercise

1. A = {−3, −1, 1, 3 }

B = {−1, 0, 1, 2, 3, 4 }

U = {x | x is an integer and −5 < x < 5}

2. C = {x | x is an integer and x is divisible by 5 }

D = {1, 3, 4}

3. E = {x | x is an integer and 0 < x < 5 }

F = {x | x is an integer and x = 2k + 1, where kis an integer}


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Exercise

1. A = {−3, −1, 1, 3 }

B = {−1, 0, 1, 2, 3, 4 }


A B = {−3, −1, 0, 1, 2, 3, 4}

A B = {−1, 1, 3}

A B = {−3}

B A = {0, 4}

A ‘ = {−4, −2, 0, 2, 4} B ‘ = {−4, −3, −2}

2. C = {x | x is an integer and x is divisible by 5 }

D = {1, 3, 4}

3. E = {x | x is an integer and 0 < x < 5 }

F = {x | x is an integer and x = 2k + 1, where kis an integer}


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Exercise


C D = {0, 1, 3, 4}

C D = { } = ∅

C D = {0} = C

D C = D

C ‘ = {−4, −3, −2, −1, 1, 2, 3, 4}

D ‘ = {−4, −3, −2, −1, 2}*Property of STI G0001

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Exercise

E F = {−

3,−

1, 1, 2, 3, 4}

E F = {1, 3}

E F = {2, 4}

F E = {−3, −1}

E ‘ = {−4, −3, −2, −1, 0}

F ‘ = {−4, −2, 0, 2, 4}



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UA \ B B \ AA ∩ B

A ∪ B

A B


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U

A

= U \ AA‘*Property of STI G0001

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A B

C*Property of STI G0001

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CheckPoint

A = {1, 2, 3, 4}

A B

13 B = {2, 4, 6, 8, 9}2468911*Property of STI G0001

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A = {3, 6, 9 . . .}

A B

0 4B = {0, 2, 4, 6 . . .}

3 6 109 13 18

Exercise


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A B

C

A = {1, 2, 3, 4}

B = {2, 3, 4, 5}

C = {3, 4, 5, 6}

1 2

34

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A B

C

A = {a, b, c}

B = {c, d, e, f}

C = {b, c, e}

ab

cd

e f


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End


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02 LCD Slide Handout 1(6)