cmsc 56 slide 06 - set theory

8/12/2019 CMSC 56 Slide 06 - Set Theory

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Computer Science 56

Discrete Mathematical Structures

in Computer Science I


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Georg Cantor 1845-1918) A Russian-born mathematician who pioneered the

study of set theory and its role in the foundation of

mathematics. He is known Father of Set Theory because of his

work.


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A Set is an unordered collection of objects. The objects in the set are also called the

elements or members of the set. Elements of the set must be:

Distinct a set should not contain duplicates.

Unordered the ordering of elements in the set isinsignificant.

Well-defined one should be able to determine if a

certain element belongs to the set.


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Elements of the set are enclosed in braces {}and each element is separated by comma.

Examples:

1. The set of vowels in English alphabet can be writtenas V = {a, e, i, o, u}

2. The sets {1, 1, 2, 3}, {1, 2, 3, 3}, and {2, 3, 1, 3} cansimply be written as {1, 2, 3}


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A set may be finite or infinite. A set is finite if its elements can be counted or

enumerated.

A set is infinite if its elements cannot be counted orenumerated.

Examples: Tell whether each set is finite or infinite.

1. Set of capital letters in English alphabet

2. Set of real numbers3. {1, 2, 3, }


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Cardinality of a set it is the number of elements in a finite set A

denoted by |A|. Example:

If A = {a, b, c, d, e}, then |A| = 5.


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Roster Method List down or enumerate the elements of the set.

This method can only be used to represent finitesets.

Rule Method give a rule which states the property satisfied by all

of the elements in the set.

It is usually in the form of {x|xhas the property}which reads, the set of all elements xsuch that xhas the property


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Example: Represent the set of all countingnumbers between 1 and 10, inclusive. Let theset be A.

1. Roster Method: A = {1, 2, 3, 4, 5, 6, 7, 8, 9 ,10} A = {1, 2, , 10}

2.

Rule Method: A = { | 1 10 and is a counting number}


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Superscript symbols such as +, -, represent positive, negative, and nonnegativevalues respectively.

Symbol Set DenotedN Set of Natural numbersZ Set of integersQ Set of rational numbersR Set of real numbersC Set of complex numbers


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Universal Set The totality of all elements under consideration.

Denoted by the symbol .

Empty Set set containing no element and is also called a null

set. Denoted by or {}. || = 0

Note {} is notan empty set, but a set containing the empty set.


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The symbol is used to denote setmembership.

A (x is an element of set A) if and onlyif x is an element of set A or x is in A. A if x is not an element of set A.


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Example: Tell whether the each of the ff. is anelement of A = {1, 2, 3, {1}}1. 42. 13. {1}4. {2}


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The symbol is used to denote setcontainment.

A B (set A is a subset of set B) if and onlyif every element of set A is an element of setB.

A B if A is not a subset of B.


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Example: Let A = {1}, B = {{1}}, C = {1, {1}}

A C and B CA B since 1 {{1}}


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Set A is said to be a proper subset of B(denoted by A B) if and only if A B andA B. Thus,

A B A B A B) Example: Let A = {1}, B = {{1}}, C = {1, {1}}

A C and B C A B since A B and A A since A = A


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Set A is equal to set B, denoted by A = B, ifand only if sets A and B have the sameelements.

Example: if A = {1, 3, 2} and B = {1, 2, 3} then A = B Also,

A = B) A B) B A)


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Let A, B and C be any set,

Remark: Every set is a subset of itself but notset is a n element of itself.

1. A The empty set is a subset ofevery set

2. A A Every set is a subset of itself3. A B A B) If A is a proper subset of B,

then A is a subset of B

4. A B) B C) A CTransitive Rules

5. A B) B C) A C


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The implicit definition of sets does not allowa set to be an element of itself since withoutthis restriction, we are led to RussellsParadox. Let R be the set of all sets that are not elements of

themselves:

R = {S | S S}


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To prove that R does no exist:a) Assume R R

By definition, since R is not an element of itself, thenR must be a member of our given set R, that is R R.

Therefore, R R implies R R, which iscontradiction.

b) Assume R R

By definition, since R R, then R must be a set that

is not an element of itself, that is R R. Therefore,

R R implies R R, which is a contradiction.


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We are led to the paradoxical conclusion thatR R if and only if R R. Hence, the set

R = {S | S S} does not exist.


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Analogy:

There was once a barber. Some say that he lived inSeville. Wherever he lived, all of the men in this

town either shaved themselves or were shaved bythe barber. And the barber only shaved the menwho did not shave themselves. Who shaves the

barber?


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A. Represent the set of multiples of 3 between 1 to20 using

1. Roster Method

2. Rule Method

B. Let A = {{x}}, B = {x, {x}}, C = {x, {x, {x}}}

Write T if the statement is true, write F if false.1. x A2. A B3. C A4. B


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Power set of a set The power set of set A, denoted by p(A), is the set

of all subsets of set A. That is,

A) = {x | x A} | A)| = 2|A|


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Example: if A = {1, 2, 3} then

A) = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3},{1, 2, 3}}

| A)| = 8


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Operation Set Representation DescriptionComplement of A A' = {x | x U x A} Set of all elementsthat are not in Abut in U

Union of A and B A B = {x | x A x B} Set of all elementsthat are in eithersets A or B

Intersection of Aand B A B = {x | x A x B} Set of all elementsthat are in both setsA and B

Difference of Aand B relativecomplement)A B = {x | x A x B}Can also be expressed as:A B = A B'

Set of all elementsthat are in set A butnot in set B

SymmetricDifference of Aand BA B = {x | (x A x B) (x A x B)}Can also be expressed as:

A B = (A B) (A B)

Set of all elementsthat are in eitherset A or B but not

both


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Example: Let A = {1, 3, 5, 7}, B = {1, 2, 3, 4}and U = {1, 2,, 10}. Determine:1. A B

2. A B

3. A B4. A'

5. B'

6. A B

7. B A


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If A B = , or A and B dont have commonelements, then they are said to be disjoint.Otherwise, A and B are said to be joint sets.


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Consider the ff sets:U = {1, 2,, 10}A = {1, 2, 3, 4, 5} C = {1, 3, 5, 7, 9}B = {2, 4, 6, 8} D = {1, 2, 9, 10}

Determine:1. A B2. D'

3. A (B - C)4. p(D)5. (A B) (A B)'


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John Venn 1834-1923) He presented a systematic development of a

method that uses geometric figures, known as VennDiagrams, in his book Symbolic Logic.

Venn Diagrams are used to analyze logicalarguments and to analyze relationships betweensets.


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Venn Diagram gives a graphicalrepresentation of sets.

Rectangle represents the universal set

Circles or any closed polygons represents thesubsets of universal set.


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Venn diagrams may also include elements orcardinality of the sets.

Example: |A| = 15, |B| = 25, |A B| = 10,

U = 35

What is |(A B)|?


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Example: Illustrate the ff sets given that:1. U = set of all UPLB students

A = set of all sophomores

B = set of all juniors

2. U = set of all UPLB students

C = set of all male students

D = set of all female students


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Venn diagrams can be used to illustrate setconcepts.

Venn diagram for some set concepts:

A

A B


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Example: Draw a Venn Diagram for each ofthe ff:1. A B

2. A B

3. A B

4. A B


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1. (A) = ADouble Negation(Involution Law)

2.(A B) = A B

(A B) = A B

De Morgans Laws

3.A B = B AA B = B A

Commutative Laws

4.A (B C) = (A B) CA (B C) = (A B) C

Associative Laws

5.A (B C) = (A B) (A C)A (B C) = (A B) (A C) Distributive Laws


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6.A A = AA A = A

Idempotency Laws

7.A = A

A U = A

Identity Laws

8.A A = UA A =

Inverse Laws

(Complement Laws)

9.A U = UA =

Domination Laws (Bound

Laws)

10.A (A B) = AA (A B) = A Absorption Laws


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Let A, B U

1. A - B = A B2. A B = (A B) - (A B)3. (A B) A (A B)4. The following statements are equivalent to

each other:

a. A Bb. A B = B

c. A B = Ad. B A

5. If A , B U then A and B are disjoint if andonly if A B = A B.

cmsc 56 slide 06 - set theory

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