MELJUN CORTES Automata Lecture Review of Important Concepts 1

8/21/2019 MELJUN CORTES Automata Lecture Review of Important Concepts 1

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Theory of Computation (With Automata Theory)

Review of Important Concepts *Property o f STIPage 1 of 15

TOPIC TITLE: Set Theory, Functions, and Relations

Specific Objectives:

At the end of the topic session, the students are expected to:

Cognit ive:

1. Explain what sets are.2. Define the different terminologies related to sets such as

subsets, infinite sets, empty sets, etc.3. Identify and correctly use the different notations of set theory.4. Apply the different operations performed on sets such as union,

intersection, complement, Cartesian product, etc.5. Use the set-builder notation to describe sets.

6. Explain how power sets are formed.7. Differentiate sets from sequences.8. Define function.9. Identify the domain and range of a given function.10. Describe a binary relation.11. Identify if a given relation is reflexive, symmetric, and transitive.12. Identify if a given relation is an equivalence function.

Affective:

1. Listen to others with respect.2. Participate in class discussions actively.

MATERIALS/EQUIPMENT:

o topic slideso OHP

TOPIC PREPARATION:

o Have the students review whatever topics were discussed duringthe previous semesters.

o Prepare the slides to be presented in class.o It is imperative for the instructor to incorporate various kinds of

teaching strategies while discussing the suggested topics. o Prepare additional examples on the topic to be presented.


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Set TheoryPage 1 of 26

Review of Set Theory

Sets

A set is a collection of objects such as numbers, letters, symbols, or other sets.These objects are called the members or elements of the set.

For example:

If set A = {0, 2, 4, 8, 16}, then the members or elements of set A are 0, 2, 4, 8,and 16.

It’s also possible to have other sets as members of sets such as B = {{1, 2}, {3, 4,5}, {6, 7, 8, 9}}. The members of set B are the sets {1, 2}, {3, 4, 5}, and {6, 7, 8,9}

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Set MembershipPage 2 of 26

In sets, the ordering and repetition of the members are not important. Thismeans that the set {1, 2, 3} is the same as the set {2, 3, 1}. As long as two setshave the same elements, they are considered the same.

Similarly, the set {1, 2, 3, 2} is the same as the set {1, 2, 3}. Since the element 2appeared twice in the first set, then one of them can be removed withoutaffecting the set.

Set Membership

The symbol is used to denote membership in a set while the symbol is usedto denote non-membership in a set.

For example:

Assume that set A = {0, 2, 4, 8, 16}. Since 8 is a member of set A, then it can bewritten as 8 A (read as “8 is a member of set A”). Similarly, since 7 is not amember of set A, then 7 A (“7 is not a member of set A”).

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SubsetsPage 3 of 26

Subsets

For two sets A and B, B is a subset of A if every member of B is also a memberof A.

Set B is a subset of set A is denoted by B A.

For example:

If set A = {0, 2, 4, 8, 16} and set B = {2, 8}, then B A since all the elements of B are in set A.

Take note that a set is a subset of itself ( A A). This is due to the fact thatevery element is present in itself.

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Infinite Set and Empty SetPage 4 of 26

Infinite Set

An infinite set is a set that has infinitely many members.

As an example, the set of integers Z = {…, -3, -2, -1, 0, 1, 2, 3, …} is an infiniteset. The set of natural numbers N = {0, 1, 2, 3, …} is also an infinite set. [Theellipses (…) indicate extension of the sequence using the pattern.]

Empty Set

An empty set (or the null set) is a set that contains no members.

If set A is an empty set, it is written as A = { } or A = .

Take note that the empty set is a subset of any other set. This is because thereare no elements in the empty set that are not in any other set.

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Union and Intersection of SetsPage 5 of 26

Union and Intersection of Sets

The union of two sets A and B is obtained by forming a single set that containsall the elements in A and B.

The union of sets A and B is written as A B.

For example:

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

Set B = {3, 4, 5, 6}

The union of sets A and B is obtained by combining the elements of both setsthereby resulting to {1, 2, 3, 4, 3, 4, 5, 6}. Since the elements 3 and 4 werewritten twice, the duplicates may be eliminated. So,

A B = {1, 2, 3, 4, 5, 6}

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Union and Intersection of Sets(continuation)Page 6 of 26

Union and Intersection of Sets (continuation)

The intersect ion of two sets A and B is obtained by forming a single set thatcontains all the elements that are in both A and B.

The intersection of sets A and B is written as A B.

For example:

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

Set B = {3, 4, 5, 6}

The intersection of sets A and B is obtained by getting the elements that arecommon to both sets thereby resulting in

A B = {3, 4}

If A B = (they have no common elements), then sets A and B are said to bedisjoint .

For example:

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

Set B = {5, 6, 7, 8}

A B =

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Complement of SetsPage 7 of 26

The Complement of Sets

Assume set A is a subset of set B ( A B). The complement of A with respectto B is the set of all elements in B that are not in A.

The complement of set A is written as Ā or A .

In this context, the bigger set (set B in this case) is referred to as the superset oruniverse.

For example:

Set A = {3, 4}Set B = {1, 2, 3, 4, 5, 6}

Since the elements 1, 2, 5, and 6 are the elements of B that are not in A, thenthe complement of set A with respect to the assumed superset B is:

A = {1, 2, 5, 6}

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Set-Builder NotationPage 8 of 26

Set-Builder Notation

The set-bui lder n otat ion is used to describe a set containing elements thathave a certain property or follow a certain rule. Instead of listing down eachelement of the set as is usually done, there are times when simply “describing” the elements of the set would be more practical.

The following designations are used in conjunction with the set-builder notation:

N = set of natural numbers

Z = set of all integers

R = set of all real numbers

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Set-Builder Notation (continuation)Page 9 of 26

Set-Builder Notation (continuation)

For example:

A = { x x Z , 10


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Power SetsPage 11 of 26

Power Sets

Recall that a set may contain other sets. With that, the power set of A is definedas the set of all subsets of set A.

The power set of A is written as P ( A) or 2 A

.

For example:

Let set A = {1, 2, 3}.

The power set may obtained by first forming all possible subsets of set A.These are:

(the empty set is a subset of any set){1}{2}{3}{1, 2}

{1, 3}{2, 3}{1, 3}{1, 2, 3} (a set is a subset of itself}

The power set of A is therefore:

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

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Sequences and TuplesPage 12 of 26

Sequences and Tuples

Recall that in sets, the ordering and repetition of the members are not important.So the set {1, 2, 3} is the same as the set {2, 3, 1}.

Sequences are similar to sets except for the fact that the ordering and repetitionof the members in sequences are important.

This means that the sequence {1, 2, 3} is different from the sequence {2, 3, 1}since the ordering in which the members are listed is significant.

Similarly, the sequence {1, 2, 3, 2} is different from the sequence {1, 2, 3}because repetition in sequences is likewise important.

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Sequences and Tuples(continuation)Page 13 of 26

Sequences and Tuples (continuation)

Like sets, sequences may be finite and infinite. Finite sequences are calledtuples .

More specifically, a sequence with k elements is a k -tuple.

For example:

The sequence {1, 2, 3} is a 3-tuple while the sequence {a, b} is a 2-tuple.

A 2-tuple is often called a pair .

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Cartesian Products of SetsPage 14 of 26

Cartesian Products of Sets

The Cartesian Produc t , or Cross Produc t of two sets A and B, is the set of all2-tuples or pairs wherein the first element of a pair is a member of set A and thesecond element of a pair is a member of set B.

The Cartesian product of sets A and B is written as A B.

Examples: Let set A = {1, 2} and set B = {x, y}

The Cartesian product of A and B ( A B) can be obtained by first forming

all possible pairs wherein the first element of a pair is a member of set A and the second element of a pair is a member of set B. These are:

{1, x}{1, y}{2, x}{2, y}

Therefore: A B = {{1, x}, {1, y}, {2, x}, {2, y}}

The Cartesian product of B and A (B A) can be obtained by first formingall possible pairs wherein the first element of a pair is a member of set B and the second element of a pair is a member of set A. These are:

{x, 1}{x, 2}{y, 1}{y, 2}

Therefore: B A = {{x, 1}, {x, 2}, {y, 1}, {y, 2}}

Note that the order when taking the Cartesian product is significant. In general, A B is not the same as B A.

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Cartesian Products of Sets(continuation)Page 15 of 26

Cartesian Products of Sets (continuation)

More examples: Let set A = {1, 2, 3} and set B = {x, y}

The Cartesian product of A and B ( A B) can be obtained by first forming

all possible pairs wherein the first element of a pair is a member of set A and the second element of a pair is a member of set B. These are:

{1, x}{1, y}{2, x}{2, y}{3, x}{3, y}

Therefore: A B = {{1, x}, {1, y}, {2, x}, {2, y}, {3, x}, {3, y}}

The Cartesian product of B and A (B A) can be obtained by first formingall possible pairs wherein the first element of a pair is a member of set B

and the second element of a pair is a member of set A. These are:

{x, 1}{x, 2}{x, 3}{y, 1}{y, 2}{y, 3}

Therefore:B A = {{x, 1}, {x, 2}, {x, 3}, {y, 1}, {y, 2}, {y, 3}}

The Cartesian product of A and A ( A A) can be obtained by first formingall possible pairs wherein the first and the second elements of a pair areboth from set A. These are:

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

Therefore: A A = {{1, 1}, {1, 2}, {1, 3}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2}, {3, 3}}

The Cartesian product of B and B (B B) can be obtained by first formingall possible pairs wherein the first and the second elements of a pair areboth from set B. These are:

{x, x}{x, y}{y, x}{y, y}

Therefore:B B = {{x, x}, {x, y}, {y, x}, {y, y}}

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Cartesian Products of Sets(continuation)Page 16 of 26

Cartesian Products of Sets (continuation)

The Cartesian product can be extended to more than two sets. Generally, theCartesian product of k sets A1, A2 , …, Ak , is the set of all k -tuples (a1, a2 , …, ak )where ai Ai .

For example: Let set A = {1, 2, 3}, set B = {x, y}, and set C = {$, %}

The Cartesian product of A, B, and C ( A B C ) can be obtained by firstforming all possible 3-tuples wherein the first element is a member of set

A, the second element of is a member of set B, and the third element ofis a member of set C . These are:

{1, x, $}{1, x, %}{1, y, $}{1, y, %}{2, x, $}{2, x, %}

{2, y, $}{2, y, %}{3, x, $}{3, x, %}{3, y, $}{3, y, %}

Therefore:

A B C = {{1, x, $}, {1, x, %}, {1, y, $}, {1, y, %}, {2, x, $}, {2, x, %},{2, y, $}, {2, y, %}, {3, x, $}, {3, x, %}, {3, y, $}, {3, y, %}}

The Cartesian product of C , B, and C (C B C ) can be obtained by firstforming all possible 3-tuples wherein the first element is a member of set C , the second element of is a member of set B, and the third element ofis a member of set C again. These are:

{$, x, $}{$, x, %}{$, y, $}{$, y, %}{%, x, $}{%, x, %}{%, y, $}{%, y, %}

Therefore:

C B C = {{$, x, $}, {$ , x, %}, {$, y, $}, {$, y, %}, {%, x, $}, {% , x, %},{%, y, $}, {%, y, %}}

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FunctionsPage 17 of 26

Functions and Relations

Functions

A funct ion is an object that defines a basic input-output relationship between

two quantities. A function accepts an input value and produces an output basedon some pre-defined rule.

Functions are usually written as f ( x ) = y . For a function f , if the input is x then theoutput is y .

Examples:

Given a function: f ( x ) = x + 1, if the input x = 2, then the output is f ( x ) = 3.

Given a function f ( x ) = x2, if the input x = 2, then the output is f ( x ) = 4.

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Functions (continuation)Page 18 of 26

Functions (continuation)

Always remember that for any given function, the same input always producesthe same output. It is possible though for two different inputs to produce thesame output.

The set of all possible values the input of a function can take on is called thedomain of the function. In other words, the domain is the set of all values thatthe function can accept as input and still operate correctly.

The set of all possible output values of a function is called the range of thefunction.

For example:

For the function f ( x ) = x + 1, the domain and range is the set of all real numbers.

For the function f ( x ) = x 2, the domain is the set of all real numbers while the

range is the set of non-negative real numbers. This is because the output is thesquare of the input and it will never be negative.

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Functions (continuation)Page 19 of 26

Functions (continuation)

A function is also called a mapping because for each input value, acorresponding output value is produced. So an output value is mapped orassociated to an input value.

For example:

For the function f ( x ) = x + 1, if x = 2, then f ( x ) = 3.

It can then be said that the function f ( x ) maps the input 2 to the output 3.The function also maps 3 to 4, 4 to 5, 5 to 6, etc.

Therefore, a function is some rule that associates to each element in its domainsome element in its range.

For example, the function f ( x ) = x 2 maps each real number to its square (2 is

mapped to 4, 3 is mapped to 9, 4 is mapped to 16, etc.)

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Binary RelationsPage 20 of 26

Binary Relations

A binary relation from set A to set B is the set of ordered pairs { x , y } where x A and y B.

The ordered pair { x , y } is in a relation R if element x is related to element y asdefined by R .

For example:

Let A = {3, 4, 5, 6, 7, 8, 9} and B = {1, 2, 3}.

Getting the Cartesian product A x B gives all possible ordered pairswhere the first element is from set A and the second element is from setB.

A x B = {{3, 1}, {3, 2}, {3, 3}, {4, 1}, {4, 2}, {4, 3}, {5, 1}, {5, 2}, {5, 3},{6, 1}, {6, 2}, {6, 3}, {7, 1}, {7, 2}, {7, 3}, {8, 1}, {8, 2}, {8, 3},{9, 1}, {9, 2}, {9, 3}}

Now if the relation R is defined as x = y 2 (this means the first element of

the ordered pair, x , is the square of the second element, y ) the relation R will be:

R = {{4, 2}, {9, 3}}

Because these are the only pairs that satisfy the relation (the firstelement is the square of the second element).

If the ordered pair { x , y } R , then we write a xRy .

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Binary Relations (continuation)Page 21 of 26

Binary Relations (continuation)

A binary relation may involve only one set, A. If that is the case, then the relationis simply a relation on set A.

For example:

Let set A = {1, 2, 3}.

Getting the Cartesian product A x B gives all possible ordered pairswhere both the first and second elements are from set A.

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

If R is defined as x


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Properties of Binary Relations(continuation)Page 23 of 26

Properties of Binary Relations (continuation)

A relation R on set A is symmetr ic if for every x, y A, xRy implies yRx . Inother words, the relation is symmetric if the two elements of all ordered pairs thatsatisfy the relation can be interchanged.

For example:

The relation “is equal to” is symmetric since x = y implies that y = x .

However, the relation “is less than” is not symmetric since 1 < 2 does notimply 2 < 1.

The relation “is a sibling of ” is symmetric since if John is a sibling ofPeter, then it implies that Peter is a sibling of John.

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A relation R on set A is t ransi t ive if for every x, y , z A, xRy and yRz implies xRz .

For example:

The relation “is less than” is transitive since if 1 < 2 and 2 < 3, then it

implies that 1 < 3.

However, the relation “is the square of ” is not transitive since 16 is thesquare of 4 and 4 is the square of 2 does not imply that 16 is the squareof 2.

The relation “is taller than” is transitive since if John is taller than Peterand Peter is taller than Paul, then it implies that John is taller than Paul.

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A binary relation R is an equiv alence relation if R is reflexive, symmetric, andtransitive.

For example:

Is the relation “looks the same as” an equivalence relation?

To determine if a relation is an equivalence relation, one must check firstif the relation is reflexive, symmetric, and transitive.

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The relation “looks the same as” is reflexive since any person looks the same ashimself (the relation can be applied to the same person).

It is symmetric since if John looks the same as Peter then it implies that Peterlooks the same as John.

It is transitive since if John looks the same as Peter and Peter looks the same asPaul, then it implies that John looks the same as Paul.

Since the relation “looks the same as” is reflexive, symmetric, and transitive, it istherefore an equivalence relation.

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MELJUN CORTES Automata Lecture Review of Important Concepts 1

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