Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكروDr. Khaled Bakro
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
Discrete Mathematics covers fundamentals of logic including propositions, truth tables, logical connectives, and propositional equivalences. A proposition is a statement that is either true or false. Logical connectives such as "and", "or", "not" are used to combine propositions into compound statements. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions. Logical equivalences show that two statements are logically equivalent even if written differently. Examples help illustrate key concepts such as tautologies, contradictions and using equivalences to prove statements.
This document introduces key concepts in propositional logic including propositions, propositional variables, truth tables, logical connectives, predicates, quantification, and translation of statements between propositional logic and English. Some key points:
- Propositions are declarative sentences that are either true or false. Propositional variables represent propositions. Logical connectives like negation, conjunction, and disjunction are used to form compound propositions.
- Truth tables define the truth values of compound propositions based on the truth values of their component propositions.
- Predicates involve variables and become propositions when values are assigned to the variables.
- Quantifiers like universal and existential quantification express the extent to which a
This document summarizes Chapter 1, Part I of a textbook on propositional logic. It introduces key concepts in propositional logic, including propositions, logical connectives like negation, conjunction, disjunction, implication, biconditional, and truth tables. It provides examples of applying propositional logic to represent English sentences, system specifications, logic puzzles and logic circuits. It also briefly describes representing knowledge about an electrical system in propositional logic for fault diagnosis in artificial intelligence.
This document defines key concepts in logic and propositions. A proposition is a declarative sentence that is either true or false. Connectives like negation, conjunction, and implication are used to build compound propositions from simpler ones. A tautology is a proposition that is always true, while a contradiction is always false. A conditional proposition in the form "if p then q" represents implication. Logical equivalence means two propositions always have the same truth value. The converse of "p implies q" is "q implies p", while the contrapositive is "not q implies not p".
The document discusses key concepts in logic including propositions, truth tables, logical connectives like conjunction and disjunction, quantifiers, and valid arguments. Some key points:
- A proposition is a statement that is either true or false.
- Truth tables define the truth values of logical connectives and conditionals.
- Quantifiers like "all" and "some" are used to make generalized statements about sets.
- Venn diagrams can represent relationships between sets graphically.
- An argument is valid if the premises necessarily make the conclusion true.
This section discusses applications of propositional logic, including translating English sentences to propositional logic, system specifications, and logic puzzles. It provides an example of translating the English sentence "You can access the Internet from campus only if you are a computer science major or you are not a freshman" to the propositional logic statement a→(c ∨ ¬f). It also gives an example of expressing the system specification "The automated reply cannot be sent when the file system is full" in propositional logic as p → ¬q.
This document provides an overview of propositional logic including:
- The basic components of propositional logic like propositions, connectives, truth tables
- Applications such as translating English sentences to logic, system specifications, puzzles
- Logical equivalences and showing equivalence through truth tables
- Sections cover propositions, connectives, truth tables, and applications including translation, specifications, puzzles
Discrete mathematics Ch2 Propositional Logic_Dr.khaled.Bakro د. خالد بكروDr. Khaled Bakro
Discrete Mathematics chapter 2 covers propositional logic. A proposition is a statement that is either true or false. Propositional logic uses propositional variables and logical operators like negation, conjunction, disjunction, implication and biconditional. Truth tables are used to determine the truth value of compound propositions formed using these operators. Logical equivalences between compound propositions can be shown using truth tables or by applying equivalence rules.
Discrete Mathematics covers fundamentals of logic including propositions, truth tables, logical connectives, and propositional equivalences. A proposition is a statement that is either true or false. Logical connectives such as "and", "or", "not" are used to combine propositions into compound statements. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions. Logical equivalences show that two statements are logically equivalent even if written differently. Examples help illustrate key concepts such as tautologies, contradictions and using equivalences to prove statements.
This document introduces key concepts in propositional logic including propositions, propositional variables, truth tables, logical connectives, predicates, quantification, and translation of statements between propositional logic and English. Some key points:
- Propositions are declarative sentences that are either true or false. Propositional variables represent propositions. Logical connectives like negation, conjunction, and disjunction are used to form compound propositions.
- Truth tables define the truth values of compound propositions based on the truth values of their component propositions.
- Predicates involve variables and become propositions when values are assigned to the variables.
- Quantifiers like universal and existential quantification express the extent to which a
This document summarizes Chapter 1, Part I of a textbook on propositional logic. It introduces key concepts in propositional logic, including propositions, logical connectives like negation, conjunction, disjunction, implication, biconditional, and truth tables. It provides examples of applying propositional logic to represent English sentences, system specifications, logic puzzles and logic circuits. It also briefly describes representing knowledge about an electrical system in propositional logic for fault diagnosis in artificial intelligence.
This document defines key concepts in logic and propositions. A proposition is a declarative sentence that is either true or false. Connectives like negation, conjunction, and implication are used to build compound propositions from simpler ones. A tautology is a proposition that is always true, while a contradiction is always false. A conditional proposition in the form "if p then q" represents implication. Logical equivalence means two propositions always have the same truth value. The converse of "p implies q" is "q implies p", while the contrapositive is "not q implies not p".
The document discusses key concepts in logic including propositions, truth tables, logical connectives like conjunction and disjunction, quantifiers, and valid arguments. Some key points:
- A proposition is a statement that is either true or false.
- Truth tables define the truth values of logical connectives and conditionals.
- Quantifiers like "all" and "some" are used to make generalized statements about sets.
- Venn diagrams can represent relationships between sets graphically.
- An argument is valid if the premises necessarily make the conclusion true.
This section discusses applications of propositional logic, including translating English sentences to propositional logic, system specifications, and logic puzzles. It provides an example of translating the English sentence "You can access the Internet from campus only if you are a computer science major or you are not a freshman" to the propositional logic statement a→(c ∨ ¬f). It also gives an example of expressing the system specification "The automated reply cannot be sent when the file system is full" in propositional logic as p → ¬q.
This document provides an overview of propositional logic including:
- The basic components of propositional logic like propositions, connectives, truth tables
- Applications such as translating English sentences to logic, system specifications, puzzles
- Logical equivalences and showing equivalence through truth tables
- Sections cover propositions, connectives, truth tables, and applications including translation, specifications, puzzles
UGC NET Computer Science & Application book.pdf [Sample]DIwakar Rajput
This document provides an overview of propositional logic and logical connectives. It defines key terms like proposition, logical connectives, truth tables, and normal forms. It describes the five basic logical connectives - negation, conjunction, disjunction, conditional, and bi-conditional. It provides truth tables and examples to explain each connective. It also discusses logical equivalences, precedence of operators, logic and bit operations, tautologies/contradictions, and normal forms. The document is a lesson on propositional logic from Diwakar Education Hub that covers basic concepts and terminology.
This document provides an overview of logic and proofs. It begins by defining logic as the study of correct reasoning, and discusses how logic is used in mathematics and computer science. Some key concepts introduced include:
- Propositions and truth values
- Logical connectives like AND, OR, NOT
- Truth tables for evaluating compound propositions
- Quantifiers like "for all" and "there exists"
- Propositional functions and their domains
- Types of proofs like direct proof, proof by contradiction, and mathematical induction
The document concludes by covering resolution proofs and the strong form of mathematical induction, which involves verifying a basis step and proving the induction step. Overall, it serves as an introduction to
The document discusses propositional logic, including:
- The basic components of propositional logic like propositions, connectives, truth tables, and logical equivalences
- Applications such as translating English sentences to propositional logic, system specifications, logic puzzles
- Representing logical relationships using truth tables and showing logical equivalences
- Using propositional logic to represent an electrical system and diagnose faults
This document summarizes Chapter 1 of the textbook "Discrete Mathematics" by R. Johnsonbaugh. It covers the topics of logic, proofs, and propositional logic. Key points include:
- Logic is the study of correct reasoning and is used in mathematics and computer science.
- A proposition is a statement that can be determined as true or false. Connectives like AND, OR, and NOT can combine propositions.
- Truth tables define the truth values of compound propositions formed from connectives.
- Quantifiers like "for all" and "there exists" are used to make universal and existential statements.
- A proof is a logical argument establishing the truth of a theorem using definitions, ax
This document provides an overview of logic and proofs in discrete mathematics. It discusses topics such as:
- Propositions and truth tables
- Logical connectives like conjunction, disjunction, negation, implication, and equivalence
- Quantifiers like universal and existential
- Different types of proofs like direct proof, proof by contradiction, and mathematical induction
- Rules of inference for propositional and quantified logic
- Resolution proofs
The document serves as an introduction to fundamental concepts in logic that are important for mathematical reasoning and proof techniques.
The document summarizes key concepts in propositional logic including:
1. Propositions are declarative sentences that are either true or false. Logical operators like negation, conjunction, disjunction, implication and biconditional are used to combine propositions.
2. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions.
3. De Morgan's laws provide equivalences for negating conjunctions and disjunctions.
The document summarizes key concepts in propositional logic including:
1. Propositions are declarative sentences that are either true or false. Logical operators like negation, conjunction, disjunction, implication and biconditional are used to combine propositions.
2. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions.
3. De Morgan's laws provide equivalences for negating conjunctions and disjunctions.
This document provides an overview of a course on discrete mathematics. The course objectives are to develop mathematical reasoning skills, learn how to analyze counting problems, and learn how to specify and analyze algorithms. Topics covered include formal logic, proofs, sets and combinatorics, algorithms and complexity, induction, counting, discrete probability, relations, graphs and trees, Boolean algebra, and finite-state and Turing machines. The primary textbook is Discrete Mathematics and Its Applications by Rosen. Chapter 1 covers propositional logic, including logical connectives like negation, conjunction, disjunction, implication, equivalence, and truth tables. Propositional equivalences like De Morgan's laws are also discussed. Bit operations on strings of Boolean values are introduced.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
1. The document discusses key concepts in propositional logic including negation, conjunction, disjunction, and exclusive disjunction.
2. Negation, conjunction, and disjunction are explained using examples and truth tables. Negation switches the truth value, conjunction requires both to be true, and disjunction requires at least one to be true.
3. Exclusive disjunction is introduced as giving true if one or the other is true but not both, as represented in its truth table.
This document provides an overview of mathematical logic. It defines key concepts such as propositions, truth values, logical connectives like negation, conjunction, disjunction, implication, biconditional, and quantifiers. Propositions are statements that can be either true or false. Logical connectives combine propositions and quantifiers specify whether statements apply to all or some cases. Truth tables are used to determine the truth values of statements combined with logical connectives. The document also discusses predicates, universal and existential quantification, and DeMorgan's laws relating negation and quantification.
Logic provides tools for analyzing reasoning and arguments through symbolic representations of propositions. A proposition is a declarative sentence that is either true or false. Logical operators such as negation, conjunction, disjunction, implication, biconditional, NAND, and NOR allow building compound propositions from simpler ones. Truth tables systematically represent the relationships between the truth values of simple and compound propositions. Key logical concepts include logical equivalence, contradiction, tautology, and logical validity.
This document discusses discrete structures and logical operators. It defines logical connectives like negation, conjunction, disjunction, exclusive or, implication, and biconditional. It provides examples of using these connectives to write compound propositions and their truth tables. It also covers translating between English statements and logical expressions. Additional topics include tautologies, contradictions, logical equivalence, De Morgan's laws, and other laws of logic. Worked examples are provided to demonstrate simplifying and proving equivalence of logical propositions.
This document summarizes Section 1.1 of a chapter on propositional logic. It introduces key concepts in propositional logic including propositions, connectives like negation, conjunction, disjunction, implication, biconditional, and truth tables. It provides examples and explanations of how to construct and interpret compound propositions using these logical connectives.
This document summarizes a lecture on discrete structures. It discusses logical equivalences, De Morgan's laws, tautologies and contradictions. It also covers laws of logic like distribution, identity and negation. Conditional propositions are defined as relating two propositions with "if-then". Truth tables are used to check logical equivalence and interpret conditionals. The contrapositive and biconditional are also introduced.
This document provides an overview of propositional logic and logical operators. It defines basic concepts like propositions, logical connectives, and truth tables. Compound propositions are formed by combining one or more propositions using logical operators like conjunction, disjunction, negation, implication, equivalence, exclusive or, and others. Computer representations of logic using bits are also discussed, where true and false map to 1 and 0, and bitwise logic operators correspond directly to the logical connectives. Precedence rules for logical operators are defined.
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptxssuser92109d
This document provides an overview of logic and mathematical concepts including:
- Definitions of statements, propositions, negation, conjunction, disjunction and their truth tables
- Logical equivalence and how to prove it using truth tables or theorems
- Conditional, biconditional, and quantified statements
- Valid and invalid arguments
- Predicates and applying quantifiers to write statements in symbolic form
- Truth conditions for universal and existential propositions
- References used to adapt the content are cited
This document defines and provides examples of basic concepts in propositional logic, including:
1. Propositions are declarative sentences that are either true or false. Common propositional variables include p, q, r.
2. Logical connectives combine propositions and include negation (¬p), conjunction (p∧q), disjunction (p∨q), conditional (p→q), biconditional (p↔q). Truth tables define the truth values of connected propositions.
3. Relations between conditionals include the converse, contrapositive, and inverse. Examples show how to derive these from a given conditional statement.
The Foundations: Logic and Proofs: Propositional Logic, Applications of Propositional Logic, Propositional Equivalence, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.
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This is the webinar recording from the June 2024 HubSpot User Group (HUG) for B2B Technology USA.
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Sign up for future HUG events at https://events.hubspot.com/b2b-technology-usa/
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UGC NET Computer Science & Application book.pdf [Sample]DIwakar Rajput
This document provides an overview of propositional logic and logical connectives. It defines key terms like proposition, logical connectives, truth tables, and normal forms. It describes the five basic logical connectives - negation, conjunction, disjunction, conditional, and bi-conditional. It provides truth tables and examples to explain each connective. It also discusses logical equivalences, precedence of operators, logic and bit operations, tautologies/contradictions, and normal forms. The document is a lesson on propositional logic from Diwakar Education Hub that covers basic concepts and terminology.
This document provides an overview of logic and proofs. It begins by defining logic as the study of correct reasoning, and discusses how logic is used in mathematics and computer science. Some key concepts introduced include:
- Propositions and truth values
- Logical connectives like AND, OR, NOT
- Truth tables for evaluating compound propositions
- Quantifiers like "for all" and "there exists"
- Propositional functions and their domains
- Types of proofs like direct proof, proof by contradiction, and mathematical induction
The document concludes by covering resolution proofs and the strong form of mathematical induction, which involves verifying a basis step and proving the induction step. Overall, it serves as an introduction to
The document discusses propositional logic, including:
- The basic components of propositional logic like propositions, connectives, truth tables, and logical equivalences
- Applications such as translating English sentences to propositional logic, system specifications, logic puzzles
- Representing logical relationships using truth tables and showing logical equivalences
- Using propositional logic to represent an electrical system and diagnose faults
This document summarizes Chapter 1 of the textbook "Discrete Mathematics" by R. Johnsonbaugh. It covers the topics of logic, proofs, and propositional logic. Key points include:
- Logic is the study of correct reasoning and is used in mathematics and computer science.
- A proposition is a statement that can be determined as true or false. Connectives like AND, OR, and NOT can combine propositions.
- Truth tables define the truth values of compound propositions formed from connectives.
- Quantifiers like "for all" and "there exists" are used to make universal and existential statements.
- A proof is a logical argument establishing the truth of a theorem using definitions, ax
This document provides an overview of logic and proofs in discrete mathematics. It discusses topics such as:
- Propositions and truth tables
- Logical connectives like conjunction, disjunction, negation, implication, and equivalence
- Quantifiers like universal and existential
- Different types of proofs like direct proof, proof by contradiction, and mathematical induction
- Rules of inference for propositional and quantified logic
- Resolution proofs
The document serves as an introduction to fundamental concepts in logic that are important for mathematical reasoning and proof techniques.
The document summarizes key concepts in propositional logic including:
1. Propositions are declarative sentences that are either true or false. Logical operators like negation, conjunction, disjunction, implication and biconditional are used to combine propositions.
2. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions.
3. De Morgan's laws provide equivalences for negating conjunctions and disjunctions.
The document summarizes key concepts in propositional logic including:
1. Propositions are declarative sentences that are either true or false. Logical operators like negation, conjunction, disjunction, implication and biconditional are used to combine propositions.
2. Truth tables are used to determine the truth value of compound propositions based on the truth values of the individual propositions.
3. De Morgan's laws provide equivalences for negating conjunctions and disjunctions.
This document provides an overview of a course on discrete mathematics. The course objectives are to develop mathematical reasoning skills, learn how to analyze counting problems, and learn how to specify and analyze algorithms. Topics covered include formal logic, proofs, sets and combinatorics, algorithms and complexity, induction, counting, discrete probability, relations, graphs and trees, Boolean algebra, and finite-state and Turing machines. The primary textbook is Discrete Mathematics and Its Applications by Rosen. Chapter 1 covers propositional logic, including logical connectives like negation, conjunction, disjunction, implication, equivalence, and truth tables. Propositional equivalences like De Morgan's laws are also discussed. Bit operations on strings of Boolean values are introduced.
The document discusses propositional logic including:
- Propositional logic uses propositions that can be either true or false and logical connectives to connect propositions.
- It introduces syntax of propositional logic including atomic and compound propositions.
- Logical connectives like negation, conjunction, disjunction, implication, and biconditional are explained along with their truth tables and significance.
- Other concepts discussed include precedence of connectives, logical equivalence, properties of operators, and limitations of propositional logic.
- Examples are provided to illustrate propositional logic concepts like truth tables, logical equivalence, and translating English statements to symbolic form.
1. The document discusses key concepts in propositional logic including negation, conjunction, disjunction, and exclusive disjunction.
2. Negation, conjunction, and disjunction are explained using examples and truth tables. Negation switches the truth value, conjunction requires both to be true, and disjunction requires at least one to be true.
3. Exclusive disjunction is introduced as giving true if one or the other is true but not both, as represented in its truth table.
This document provides an overview of mathematical logic. It defines key concepts such as propositions, truth values, logical connectives like negation, conjunction, disjunction, implication, biconditional, and quantifiers. Propositions are statements that can be either true or false. Logical connectives combine propositions and quantifiers specify whether statements apply to all or some cases. Truth tables are used to determine the truth values of statements combined with logical connectives. The document also discusses predicates, universal and existential quantification, and DeMorgan's laws relating negation and quantification.
Logic provides tools for analyzing reasoning and arguments through symbolic representations of propositions. A proposition is a declarative sentence that is either true or false. Logical operators such as negation, conjunction, disjunction, implication, biconditional, NAND, and NOR allow building compound propositions from simpler ones. Truth tables systematically represent the relationships between the truth values of simple and compound propositions. Key logical concepts include logical equivalence, contradiction, tautology, and logical validity.
This document discusses discrete structures and logical operators. It defines logical connectives like negation, conjunction, disjunction, exclusive or, implication, and biconditional. It provides examples of using these connectives to write compound propositions and their truth tables. It also covers translating between English statements and logical expressions. Additional topics include tautologies, contradictions, logical equivalence, De Morgan's laws, and other laws of logic. Worked examples are provided to demonstrate simplifying and proving equivalence of logical propositions.
This document summarizes Section 1.1 of a chapter on propositional logic. It introduces key concepts in propositional logic including propositions, connectives like negation, conjunction, disjunction, implication, biconditional, and truth tables. It provides examples and explanations of how to construct and interpret compound propositions using these logical connectives.
This document summarizes a lecture on discrete structures. It discusses logical equivalences, De Morgan's laws, tautologies and contradictions. It also covers laws of logic like distribution, identity and negation. Conditional propositions are defined as relating two propositions with "if-then". Truth tables are used to check logical equivalence and interpret conditionals. The contrapositive and biconditional are also introduced.
This document provides an overview of propositional logic and logical operators. It defines basic concepts like propositions, logical connectives, and truth tables. Compound propositions are formed by combining one or more propositions using logical operators like conjunction, disjunction, negation, implication, equivalence, exclusive or, and others. Computer representations of logic using bits are also discussed, where true and false map to 1 and 0, and bitwise logic operators correspond directly to the logical connectives. Precedence rules for logical operators are defined.
20220818151924_PPT01 - The Logic of Compound and Quantitative Statement.pptxssuser92109d
This document provides an overview of logic and mathematical concepts including:
- Definitions of statements, propositions, negation, conjunction, disjunction and their truth tables
- Logical equivalence and how to prove it using truth tables or theorems
- Conditional, biconditional, and quantified statements
- Valid and invalid arguments
- Predicates and applying quantifiers to write statements in symbolic form
- Truth conditions for universal and existential propositions
- References used to adapt the content are cited
This document defines and provides examples of basic concepts in propositional logic, including:
1. Propositions are declarative sentences that are either true or false. Common propositional variables include p, q, r.
2. Logical connectives combine propositions and include negation (¬p), conjunction (p∧q), disjunction (p∨q), conditional (p→q), biconditional (p↔q). Truth tables define the truth values of connected propositions.
3. Relations between conditionals include the converse, contrapositive, and inverse. Examples show how to derive these from a given conditional statement.
The Foundations: Logic and Proofs: Propositional Logic, Applications of Propositional Logic, Propositional Equivalence, Predicates and Quantifiers, Nested Quantifiers, Rules of Inference, Introduction to Proofs, Proof Methods and Strategy.
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Sign up for future HUG events at https://events.hubspot.com/b2b-technology-usa/
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lectures in prolog in order to advance in artificial intelligence
1. Chapter 1: The Foundations:
Logic and Proofs
Discrete Mathematics and Its Applications
Lingma Acheson (linglu@iupui.edu)
Department of Computer and Information Science, IUPUI
1
2. 1.1 Propositional Logic
A proposition is a declarative sentence (a
sentence that declares a fact) that is either
true or false, but not both.
Are the following sentences propositions?
Toronto is the capital of Canada.
Read this carefully.
1+2=3
x+1=2
What time is it?
(No)
(No)
(No)
(Yes)
(Yes)
Introduction
2
3. 1.1 Propositional Logic
Propositional Logic – the area of logic that
deals with propositions
Propositional Variables – variables that
represent propositions: p, q, r, s
E.g. Proposition p – “Today is Friday.”
Truth values – T, F
3
4. 1.1 Propositional Logic
Examples
Find the negation of the proposition “Today is Friday.” and
express this in simple English.
Find the negation of the proposition “At least 10 inches of rain
fell today in Miami.” and express this in simple English.
DEFINITION 1
Let p be a proposition. The negation of p, denoted by ¬p, is the statement
“It is not the case that p.”
The proposition ¬p is read “not p.” The truth value of the negation of p, ¬p
is the opposite of the truth value of p.
Solution: The negation is “It is not the case that today is Friday.”
In simple English, “Today is not Friday.” or “It is not
Friday today.”
Solution: The negation is “It is not the case that at least 10 inches
of rain fell today in Miami.”
In simple English, “Less than 10 inches of rain fell today
in Miami.”
4
5. 1.1 Propositional Logic
Note: Always assume fixed times, fixed places, and particular people
unless otherwise noted.
Truth table:
Logical operators are used to form new propositions from two or more
existing propositions. The logical operators are also called
connectives.
The Truth Table for the
Negation of a Proposition.
p ¬p
T
F
F
T
5
6. 1.1 Propositional Logic
Examples
Find the conjunction of the propositions p and q where p is the
proposition “Today is Friday.” and q is the proposition “It is
raining today.”, and the truth value of the conjunction.
DEFINITION 2
Let p and q be propositions. The conjunction of p and q, denoted by p
Λ q, is the proposition “p and q”. The conjunction p Λ q is true when
both p and q are true and is false otherwise.
Solution: The conjunction is the proposition “Today is Friday and it
is raining today.” The proposition is true on rainy Fridays.
6
7. 1.1 Propositional Logic
Note:
inclusive or : The disjunction is true when at least one of the two
propositions is true.
E.g. “Students who have taken calculus or computer science can take
this class.” – those who take one or both classes.
exclusive or : The disjunction is true only when one of the
proposition is true.
E.g. “Students who have taken calculus or computer science, but not
both, can take this class.” – only those who take one of them.
Definition 3 uses inclusive or.
DEFINITION 3
Let p and q be propositions. The disjunction of p and q, denoted by p ν
q, is the proposition “p or q”. The conjunction p ν q is false when both
p and q are false and is true otherwise.
7
8. 1.1 Propositional Logic
The Truth Table for
the Conjunction of
Two Propositions.
p q p Λ q
T T
T F
F T
F F
T
F
F
F
The Truth Table for
the Disjunction of
Two Propositions.
p q p ν q
T T
T F
F T
F F
T
T
T
F
DEFINITION 4
Let p and q be propositions. The exclusive or of p and q, denoted by p q,
is the proposition that is true when exactly one of p and q is true and is
false otherwise.
The Truth Table for the
Exclusive Or (XOR) of
Two Propositions.
p q p q
T T
T F
F T
F F
F
T
T
F
8
9. 1.1 Propositional Logic
DEFINITION 5
Let p and q be propositions. The conditional statement p → q, is the
proposition “if p, then q.” The conditional statement is false when p is
true and q is false, and true otherwise. In the conditional statement p
→ q, p is called the hypothesis (or antecedent or premise) and q is
called the conclusion (or consequence).
Conditional Statements
A conditional statement is also called an implication.
Example: “If I am elected, then I will lower taxes.” p → q
implication:
elected, lower taxes. T T | T
not elected, lower taxes. F T | T
not elected, not lower taxes. F F | T
elected, not lower taxes. T F | F
9
10. 1.1 Propositional Logic
Example:
Let p be the statement “Maria learns discrete mathematics.” and
q the statement “Maria will find a good job.” Express the
statement p → q as a statement in English.
Solution: Any of the following -
“If Maria learns discrete mathematics, then she will find a
good job.
“Maria will find a good job when she learns discrete
mathematics.”
“For Maria to get a good job, it is sufficient for her to
learn discrete mathematics.”
“Maria will find a good job unless she does not learn
discrete mathematics.”
10
11. 1.1 Propositional Logic
Other conditional statements:
Converse of p → q : q → p
Contrapositive of p → q : ¬ q → ¬ p
Inverse of p → q : ¬ p → ¬ q
11
12. 1.1 Propositional Logic
p ↔ q has the same truth value as (p → q) Λ (q → p)
“if and only if” can be expressed by “iff”
Example:
Let p be the statement “You can take the flight” and let q be the
statement “You buy a ticket.” Then p ↔ q is the statement
“You can take the flight if and only if you buy a ticket.”
Implication:
If you buy a ticket you can take the flight.
If you don’t buy a ticket you cannot take the flight.
DEFINITION 6
Let p and q be propositions. The biconditional statement p ↔ q is the
proposition “p if and only if q.” The biconditional statement p ↔ q is
true when p and q have the same truth values, and is false otherwise.
Biconditional statements are also called bi-implications.
12
14. 1.1 Propositional Logic
We can use connectives to build up complicated compound
propositions involving any number of propositional variables, then
use truth tables to determine the truth value of these compound
propositions.
Example: Construct the truth table of the compound proposition
(p ν ¬q) → (p Λ q).
Truth Tables of Compound Propositions
The Truth Table of (p ν ¬q) → (p Λ q).
p q ¬q p ν ¬q p Λ q (p ν ¬q) → (p Λ q)
T T
T F
F T
F F
F
T
F
T
T
T
F
T
T
F
F
F
T
F
T
F 14
15. 1.1 Propositional Logic
We can use parentheses to specify the order in which logical
operators in a compound proposition are to be applied.
To reduce the number of parentheses, the precedence order is
defined for logical operators.
Precedence of Logical Operators
Precedence of Logical Operators.
Operator Precedence
¬ 1
Λ
ν
2
3
→
↔
4
5
E.g. ¬p Λ q = (¬p ) Λ q
p Λ q ν r = (p Λ q ) ν r
p ν q Λ r = p ν (q Λ r)
15
16. 1.1 Propositional Logic
English (and every other human language) is often ambiguous.
Translating sentences into compound statements removes the
ambiguity.
Example: How can this English sentence be translated into a logical
expression?
“You cannot ride the roller coaster if you are under 4 feet
tall unless you are older than 16 years old.”
Translating English Sentences
Solution: Let q, r, and s represent “You can ride the roller coaster,”
“You are under 4 feet tall,” and “You are older than
16 years old.” The sentence can be translated into:
(r Λ ¬ s) → ¬q.
16
17. 1.1 Propositional Logic
Example: How can this English sentence be translated into a logical
expression?
“You can access the Internet from campus only if you are a
computer science major or you are not a freshman.”
Solution: Let a, c, and f represent “You can access the Internet from
campus,” “You are a computer science major,” and “You are
a freshman.” The sentence can be translated into:
a → (c ν ¬f).
17
18. 1.1 Propositional Logic
Computers represent information using bits.
A bit is a symbol with two possible values, 0 and 1.
By convention, 1 represents T (true) and 0 represents F (false).
A variable is called a Boolean variable if its value is either true or
false.
Bit operation – replace true by 1 and false by 0 in logical
operations.
Table for the Bit Operators OR, AND, and XOR.
x y x ν y x Λ y x y
0
0
1
1
0
1
0
1
0
1
1
1
0
0
0
1
0
1
1
0
Logic and Bit Operations
18
19. 1.1 Propositional Logic
Example: Find the bitwise OR, bitwise AND, and bitwise XOR of the
bit string 01 1011 0110 and 11 0001 1101.
DEFINITION 7
A bit string is a sequence of zero or more bits. The length of this string
is the number of bits in the string.
Solution:
01 1011 0110
11 0001 1101
-------------------
11 1011 1111 bitwise OR
01 0001 0100 bitwise AND
10 1010 1011 bitwise XOR
19
20. 1.2 Propositional Equivalences
DEFINITION 1
A compound proposition that is always true, no matter what the truth
values of the propositions that occurs in it, is called a tautology. A
compound proposition that is always false is called a contradiction. A
compound proposition that is neither a tautology or a contradiction is
called a contingency.
Introduction
Examples of a Tautology and a Contradiction.
p ¬p p ν ¬p p Λ ¬p
T
F
F
T
T
T
F
F
20
21. 1.2 Propositional Equivalences
DEFINITION 2
The compound propositions p and q are called logically equivalent if p ↔
q is a tautology. The notation p ≡ q denotes that p and q are logically
equivalent.
Logical Equivalences
Truth Tables for ¬p ν q and p → q .
p q ¬p ¬p ν q p → q
T
T
F
F
T
F
T
F
F
F
T
T
T
F
T
T
T
F
T
T
Compound propositions that have the same truth values in all
possible cases are called logically equivalent.
Example: Show that ¬p ν q and p → q are logically equivalent.
21
22. 1.2 Propositional Equivalences
In general, 2n rows are required if a compound proposition involves n
propositional variables in order to get the combination of all truth
values.
See page 24, 25 for more logical equivalences.
22
23. 1.2 Propositional Equivalences
Constructing New Logical Equivalences
Example: Show that ¬(p → q ) and p Λ ¬q are logically equivalent.
Solution:
¬(p → q ) ≡ ¬(¬p ν q) by example on slide 21
≡ ¬(¬p) Λ ¬q by the second De Morgan law
≡ p Λ ¬q by the double negation law
Example: Show that (p Λ q) → (p ν q) is a tautology.
Solution: To show that this statement is a tautology, we will use logical
equivalences to demonstrate that it is logically equivalent to T.
(p Λ q) → (p ν q) ≡ ¬(p Λ q) ν (p ν q) by example on slide 21
≡ (¬ p ν ¬q) ν (p ν q) by the first De Morgan law
≡ (¬ p ν p) ν (¬ q ν q) by the associative and
communicative law for disjunction
≡ T ν T
≡ T
Note: The above examples can also be done using truth tables.
23