#2 formal methods – principles of logic

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Prepared by: Sharif Omar Salem – [email protected]

Formal Methods:Introduction in Logic

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Historical ViewSet Theory Symbolic Logic

Truth Tables

This lectute content

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Philosophical Logic 500 BC (Before Christ) to 19th Century

Symbolic LogicMid to late 19th Century

Mathematical LogicLate 19th to mid 20th Century

Logic in Computer Science

Historical view

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500 B.C – 19th CenturyLogic dealt with arguments in the natural language used

by humans.Example

All men are moral.Socrates is a manTherefore, Socrates is moral.

Historical view -Philosophical Logic

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Natural language is very ambiguous.Eric does not believe that Mary can pass any test.I only borrowed your car.Tom hates Jim and he likes Mary.

It led to many paradoxes.“This sentence is a lie.” (The Liar’s Paradox)

Historical view - Philosophical Logic

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Mid to late 19th Century.Attempted to formulate logic in terms of a mathematical

languageRules of inference were modeled after various laws for

manipulating algebraic expressions.

Historical view - Symbolic Logic

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Late 19th to mid 20th CenturyFrege proposed logic as a language for mathematics in

1879.With the rigor of this new foundation, Cantor was able to

analyze the notion of infinity in ways that were previously impossible. (2N is strictly larger than N)

Russell’s ParadoxT = { S | S S}∉

Historical view - Mathematical Logic

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In computer science, we design and study systems through the use of formal languages that can themselves be interpreted by a formal system.Boolean circuitsProgramming languagesDesign Validation and verificationAI, Security. Etc.

Historical view- Logic in Computer Science

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Propositional LogicFirst Order LogicHigher Order LogicTheory of ConstructionReal-time Logic, Temporal LogicProcess Algebras Linear Logic

Historical view - Logics in Computer Science

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Understanding set theory helps people to …

see things in terms of systems

organize things into groups

begin to understand logic

Set Theory - Why Study Set Theory?

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These mathematicians influenced the development of set theory and logic:

Georg Cantor John Venn George Boole Augustus DeMorgan

Set Theory - Key Mathematicians

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developed set theoryset theory was not initially accepted

because it was radically differentset theory today is widely accepted and is

used in many areas of mathematics

Set Theory Georg Cantor 1845 -1918

the concept of infinity was expanded by Cantor’s set theoryCantor proved there are “levels of infinity”an infinitude of integers initially ending with or an infinitude of real numbers exist between 1 and 2;there are more real numbers than there are integers…

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John Venn 1834-1923studied and taught logic and probability theoryarticulated Boole’s algebra of logicdevised a simple way to diagram set operations

(Venn Diagrams)

Set Theory

George Boole 1815-1864self‑taught mathematician with an interest in logicdeveloped an algebra of logic (Boolean Algebra)featured the operators

andornotnor (exclusive or)

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developed two laws of negationinterested, like other

mathematicians, in using mathematics to demonstrate logic

furthered Boole’s work of incorporating logic and mathematics

formally stated the laws of set theory

Set Theory Augustus De Morgan 1806-1871

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A set is a collection of elementsAn element is an object contained in a setIf every element of Set A is also contained in Set B, then Set A is

a subset of Set BA is a proper subset of B if B has more elements than A does

The universal set contains all of the elements relevant to a given discussion

Set TheoryBasic Set Theory Definitions

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the universal set is a deck of ordinary playing cards

each card is an element in the universal set

some subsets are:face cardsnumbered cardssuitspoker hands

Set Theory Simple Set Example

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Symbol Meaning

Upper case designates set nameLower case designates set elements{ } enclose elements in set or is (or is not) an element of is a subset of (includes equal sets) is a proper subset of is not a subset of is a superset of| or : such that (if a condition is true)| | the cardinality of a set

Set Theory Set Theory Notation

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a set is a collection of objects

sets can be defined two ways:by listing each elementby defining the rules for membership

Examples:A = {2,4,6,8,10}A = {x | x is a positive even integer <12}

Set Theory Set Notation: Defining Sets

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an element is a member of a setnotation: means “is an element of”

means “is not an element of”Examples:

A = {1, 2, 3, 4}

1 A 6 A 2 A z AB = {x | x is an even number 10}

2 B 9 B 4 B z B

Set Theory Set Notation Elements

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a subset exists when a set’s members are also contained in another set

notation:

means “is a subset of”

means “is a proper subset of”

means “is not a subset of”

Set Theory Subsets

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A = {x | x is a positive integer 8}set A contains: 1, 2, 3, 4, 5, 6, 7, 8

B = {x | x is a positive even integer 10}set B contains: 2, 4, 6, 8

C = {2, 4, 6, 8, 10}set C contains: 2, 4, 6, 8, 10

Subset RelationshipsA A A B A CB A B B B CC A C B C C

Set Theory Subset Relationships

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Two sets are equal if and only if they contain precisely the same elements.

The order in which the elements are listed is unimportant.Elements may be repeated in set definitions without increasing

the size of the sets.Examples:

A = {1, 2, 3, 4} B = {1, 4, 2, 3}A B and B A; therefore, A = B and B = A

A = {1, 2, 2, 3, 4, 1, 2} B = {1, 2, 3, 4}A B and B A; therefore, A = B and B = A

Set Theory Set Equality

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Cardinality refers to the number of elements in a setA finite set has a countable number of elementsAn infinite set has at least as many elements as the set of

natural numbersnotation: |A| represents the cardinality of Set A

Set Definition CardinalityA = {x | x is a lower case letter} |A| = 26B = {2, 3, 4, 5, 6, 7} |B| = 6A = {1, 2, 3, …} |A| =

Set Theory Cardinality of Sets

0

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The universal set is the set of all things pertinent to to a given discussionand is designated by the symbol U

Example:U = {all students at LUCT}Some Subsets:

A = {all Computer Technology students}B = {freshmen students}C = {sophomore students}

Set Theory Universal Sets

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Any set that contains no elements is called the empty setthe empty set is a subset of every set including itselfnotation: { } or

Examples ~ both A and B are emptyA = {x | x is a Proton Civic car}B = {x | x is a positive number 0}

Set Theory The Empty Set

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The power set is the set of all subsets that can be created from a given set

The cardinality of the power set is 2 to the power of the given set’s cardinality

notation: P (set name)Example:A = {a, b, c} where |A| = 3P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, }

and |P (A)| = 8In general, if |A| = n, then |P (A) | = 2n

Set Theory The Power Set ( P )

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Z represents the set of integers Z+ is the set of positive integers andZ- is the set of negative integers

N represents the set of natural numbers

ℝ represents the set of real numbers

Q represents the set of rational numbers

Set Theory Special Sets

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Venn diagrams show relationships between sets and their elements

Set Theory Venn Diagrams

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Set Definition ElementsA = {x | x Z+ and x 8} 1 2 3 4 5 6 7 8B = {x | x Z+; x is even and 10} 2 4 6 8 10

A BB A

Set Theory Venn Diagram Example 1

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Set Definition ElementsA = {x | x Z+ and x 9} 1 2 3 4 5 6 7 8 9B = {x | x Z+ ; x is even and 8} 2 4 6 8

A BB AA B


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Set Definition U = {1, 2, 3, 4, 5, 6, 7, 8}

A = {1, 2, 6, 7}B = {2, 3, 4, 7}C = {4, 5, 6, 7}


A = {1, 2, 6, 7}

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“A union B” is the set of all elements that are in A, or B, or both.

This is similar to the logical “or” operator.

Set Theory Combining Sets – Set Union

A B

A B

A B

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“A intersect B” is the set of all elements that are in both A and B.

This is similar to the logical “and”

Set Theory Combining Sets – Set Intersection

A B

A B

A B

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“A complement,” or “not A” is the set of all elements not in A.

The complement operator is similar to the logical not, and is reflexive, that is,

Set Theory Set Complement

A

A A

A A

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The set difference “A minus B” is the set of elements that are in A, with those that are in B subtracted out. Another way of putting it is, it is the set of elements that are in A, and not in B, so

Set TheorySet Difference

A B

A B A B

A B A B

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Set TheoryExamples

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

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

{1,2,3,4,5,6}

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

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Set TheorySome test Questions

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Symbolic logic is a collection of languages that use symbols to represent facts,

events, and actions,and provide rules to symbolize reasoning.

Given the specification of a system and a collection of desirable properties, both written in logic formulas, we can attempt to prove that these desirable properties are logical consequences of the specification.

Symbolic Logic

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Definition of a statement:A statement, also called a proposition, is a sentence that is either true or false, but not both.

Hence the truth value of a statement is T (1) or F (0)

Examples: Which ones are statements?All mathematicians wear sandals.5 is greater than –2.Where do you live?You are a cool person.Anyone who wears sandals is an algebraist.

Symbolic LogicStatement

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An example to illustrate how logic really helps us (3 statements written below): All LUCT staff wear Black. Anyone who wears black is an employee. Therefore, all LUCT staff are employees.

Logic is of no help in determining the individual truth of these statements.

Logic helps in concluding fact from another facts (reasoning) However, if the first two statements are true, logic assures the truth of the

third statement.

Logical methods are used in mathematics to prove theorems and in computer science to prove that programs do what they are supposed to do.

Symbolic LogicStatements and Logic

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Usually, letters like A, B, C, D, etc. are used to represent statements.

Logical connectives are symbols such as

Λ , V , , , ’Λ represents and, V represents or, represents equivalence,

represents implication, ‘ represents negation.

A statement form or propositional form is an expression made up of statement variables (such as A, B, and C) and logical connectives (such as Λ, V, , )

Example: (A V B) (B Λ C)

Symbolic LogicStatements and Logical Connectives

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Connective # 1: Conjunction “AND” (symbol Λ)

Symbol Unicode is 8743If A and B are statement variables, the conjunction of A and B is A Λ B,

which is read “A and B”.A Λ B is true when both A and B are true. A Λ B is false when at least one of A or B is false. A and B are called the conjuncts of A Λ B.

Symbolic LogicDefinitions for Logical Connectives

A B A Λ B

T T T

T F F

F T F

F F F

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Connective # 2: Disjunction “OR” (symbol V)

Symbol Unicode is 8744If A and B are statement variables,

the disjunction of A and B is A V B, which is read “A or B”.A V B is true when at least one of A or B is true. A V B is false when both A and B are false.


A B A V B

T T T

T F T

F T T

F F F

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Connective # 3: Implication (symbol )

Symbol Unicode is 8594If A and B are statement variables, the symbolic form of “if A then B” is

A B. This may also be read “A implies B” or “A only if B.”“If A then B” is false when A is true and B is false, and it is true

otherwise. Note: A B is true if A is false, regardless of the truth of B


A B AB

T T T

T F F

F T T

F F T

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Connective # 4: Equivalence (symbol )

Symbol Unicode is 8596 If A and B are statement variables, the symbolic form of “A if, and only if, B”

and is denoted A B. It is true if both A and B have the same truth values. It is false if A and B have opposite truth values. The truth table is as follows:Note: A B is a short form for (A B) Λ (B A)


A B AB BA (A B) Λ (B A)

T T T T T

T F F T F

F T T F F

F F T T T

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Connective #5: Negation (symbol )

Symbol Unicode is 0732If A is a statement variable, the negation of A is “not A” and is denoted

A.It has the opposite truth value from A: if A is true, then A is false; if A

is false, then A is true. Example of a negation:

A: 5 is greater than –2 A : 5 is less than –2B: She likes butter B : She dislikes butter / She hates butter

Sometimes we use the symbol ¬ for negation


A AT FF T

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A truth table is a table that displays the truth values of a statement form which correspond to the different combinations of truth values for the variables.

Symbolic LogicTruth Tables

A AT FF T

A B A Λ B

T T TT F FF T FF F F

A B A V B

T T TT F TF T TF F F

A B ABT T TT F FF T TF F T

A B ABT T TT F FF T FF F T

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Combining letters, connectives, and parentheses can generate an expression which is meaningful, called a wff.e.g. (A B) V (B A) is a wff but A )) V B ( C) is not

To reduce the number of parentheses, an order is stipulated in which the connectives can be applied, called the order of precedence, which is as follows:

Connectives within innermost parentheses first and then progress outwardsNegation ()Conjunction (Λ), Disjunction (V)Implication () Equivalence ()

Hence, A V B C is the same as (A V B) C

Symbolic LogicWell Formed Formula (wff)

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The truth table for the wff A V B (A V B) shown below. The main connective, according to the rules of precedence, is implication.

Symbolic LogicTruth Tables for some wffs

A B B A V B A V B (A V B)

A V B (A V B)

T T F T T F F

T F T T T F F

F T F F T F T

F F T T F T T

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Definition of tautology (VALIDITY): A wff that is intrinsically true, i.e. no matter what the truth value of

the statements that comprise the wff. e.g. It will rain today or it will not rain today ( A V A )P Q where P is A B and Q is A V B

Definition of a contradiction (Unsatisfy): A wff that is intrinsically false, i.e. no matter what the truth value of

the statements that comprise the wff. e.g. It will rain today and it will not rain today ( A Λ A )(A Λ B) Λ A

Usually, tautology is represented by 1 and contradiction by 0

Symbolic LogicTautology and Contradiction

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Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables.

The logical equivalence of statement forms P and Q is denoted by writing P Q or P Q.

Truth table for (A V B) V C A V (B V C)

Symbolic LogicTautological Equivalences

A B C A V B B V C (A V B) V C

A V (B V C)

T T T T T T TT T F T T T TT F T T T T TT F F T F T TF T T T T T TF T F T T T TF F T F T T TF F F F F F F

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The equivalences are listed in pairs, hence they are called duals of each other.

One equivalence can be obtained from another by replacing V with Λ and 0 with 1 or vice versa.

Prove the distributive property using truth tables.

Symbolic LogicSome Common Equivalences

Commutative

A V B B V A A Λ B B Λ A

Associative

(A V B) V C A V (B V C)

(A Λ B) Λ C A Λ (B Λ C)

Distributive

A V (B Λ C) (A V B) Λ (A V C)

A Λ (B V C) (A Λ B) V (A Λ C)

Identity A V 0 A A Λ 1 A

Complement

A V A 1 A Λ A 0

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1. (A V B) A Λ B

2. (A Λ B) A V B

Symbolic LogicDe Morgan’s Laws

A B A B A V B (A V B)

A Λ B

T T F F T F FT F F T T F FF T T F T F FF F T T F T T

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End of Lecture

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Next LecturePropositional Logic

#2 formal methods – principles of logic

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Transcript of #2 formal methods – principles of logic