This document discusses predicates and quantifiers. It defines universal and existential quantifiers, as well as the uniqueness quantifier. It also covers precedence of quantifiers, logical equivalences involving quantifiers, and De Morgan's laws for quantifiers. Examples are provided to demonstrate how to negate quantified expressions and use De Morgan's laws to determine logical equivalences.
The document defines logical quantifiers such as existence and uniqueness quantifiers. It discusses how quantifiers can be used to restrict domains and bind variables. It provides examples of translating English statements to logical expressions using quantifiers and discusses precedence, logical equivalences, and negating quantifier expressions.
This document discusses quantification in logic. Quantification transforms a propositional function into a proposition by expressing the extent to which a predicate is true. There are two main types of quantification: universal quantification and existential quantification. Universal quantification expresses that a predicate is true for every element, while existential quantification expresses that a predicate is true for at least one element. The document provides examples and pros and cons of each type of quantification and notes that quantification operators like ∀ and ∃ take precedence over logical operators.
This lecture covers propositional equivalences like tautology and contradiction, logical equivalences that have the same truth values, De Morgan's law, and predicates and quantifiers. Predicates assign properties to variables, and quantifiers like the universal and existential quantifier specify whether a property holds for all or some variables. The lecture also discusses binding variables and nested quantified expressions.
This document presents an overview of first order ordinary differential equations and applications. It contains:
1) The standard form of a linear first order differential equation and examples of solving three types of equations.
2) Applications of differential equations to model population growth and finding the equation of a curve given its slope at a point.
3) Solutions to the examples and applications in 3 sentences or less.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
This document presents an introduction to rules of inference. It defines an argument and valid argument. It then explains several common rules of inference like modus ponens, modus tollens, addition, and simplification. Modus ponens and modus tollens are based on tautologies that make the conclusions logically follow from the premises. It also discusses two common fallacies - affirming the conclusion and denying the hypothesis - which are not valid rules of inference because they are not based on tautologies. Examples are provided to illustrate each rule of inference and fallacy.
The document defines logical quantifiers such as existence and uniqueness quantifiers. It discusses how quantifiers can be used to restrict domains and bind variables. It provides examples of translating English statements to logical expressions using quantifiers and discusses precedence, logical equivalences, and negating quantifier expressions.
This document discusses quantification in logic. Quantification transforms a propositional function into a proposition by expressing the extent to which a predicate is true. There are two main types of quantification: universal quantification and existential quantification. Universal quantification expresses that a predicate is true for every element, while existential quantification expresses that a predicate is true for at least one element. The document provides examples and pros and cons of each type of quantification and notes that quantification operators like ∀ and ∃ take precedence over logical operators.
This lecture covers propositional equivalences like tautology and contradiction, logical equivalences that have the same truth values, De Morgan's law, and predicates and quantifiers. Predicates assign properties to variables, and quantifiers like the universal and existential quantifier specify whether a property holds for all or some variables. The lecture also discusses binding variables and nested quantified expressions.
This document presents an overview of first order ordinary differential equations and applications. It contains:
1) The standard form of a linear first order differential equation and examples of solving three types of equations.
2) Applications of differential equations to model population growth and finding the equation of a curve given its slope at a point.
3) Solutions to the examples and applications in 3 sentences or less.
Lecture 2 predicates quantifiers and rules of inferenceasimnawaz54
1) Predicates become propositions when variables are quantified by assigning values or using quantifiers. Quantifiers like ∀ and ∃ are used to make statements true or false for all or some values.
2) ∀ (universal quantifier) means "for all" and makes a statement true for all values of a variable. ∃ (existential quantifier) means "there exists" and makes a statement true if it is true for at least one value.
3) Predicates with unbound variables are neither true nor false. Binding variables by assigning values or using quantifiers turns predicates into propositions that can be evaluated as true or false.
This document presents an introduction to rules of inference. It defines an argument and valid argument. It then explains several common rules of inference like modus ponens, modus tollens, addition, and simplification. Modus ponens and modus tollens are based on tautologies that make the conclusions logically follow from the premises. It also discusses two common fallacies - affirming the conclusion and denying the hypothesis - which are not valid rules of inference because they are not based on tautologies. Examples are provided to illustrate each rule of inference and fallacy.
The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
This document discusses logical expressions and quantifiers. It provides examples of using quantifiers like ∀ and ∃ to represent statements involving sets, predicates and relationships between variables. It also discusses the meanings of multiple quantifiers, bound and free variables, and how to translate statements into logical expressions.
This document provides an introduction to the Master Theorem, which can be used to determine the asymptotic runtime of recursive algorithms. It presents the three main conditions of the Master Theorem and examples of applying it to solve recurrence relations. It also notes some pitfalls in using the Master Theorem and briefly introduces a fourth condition for cases where the non-recursive term is polylogarithmic rather than polynomial.
Discrete Math Chapter 1 :The Foundations: Logic and ProofsAmr Rashed
The document describes Chapter 1 of a textbook on discrete mathematics and its applications. Chapter 1 covers propositional logic, propositional equivalences, predicates and quantifiers, and nested quantifiers. It defines basic concepts such as propositional variables, logical operators, truth tables, logical equivalence, predicates, quantifiers, and translating between logical expressions and English sentences. Examples are provided to illustrate different logical equivalences and how to negate quantified statements. The chapter introduces key foundations of logic and proofs that are important for discrete mathematics.
First order linear differential equationNofal Umair
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
This document discusses various methods of mathematical proof, including:
1. Direct proofs, which are used to prove statements of the form "If P then Q" by listing statements from P to Q using axioms and inference rules.
2. Proof by contraposition, which proves "If P then Q" by showing "If not Q then not P".
3. Proof by contradiction, which assumes the negation of what is to be proved and arrives at a contradiction.
The document discusses the Method of Frobenius for solving ordinary differential equations (ODEs) with singular points. It states that the solution for such an ODE is given as an infinite series involving powers of x. To determine the coefficients in the series, one substitutes the series solution into the original ODE, equates coefficients of like powers of x, and obtains the indical equation. Solving this indical equation gives the indicial solution and recurrence relations for the coefficients.
The document is an introduction to ordinary differential equations prepared by Ahmed Haider Ahmed. It defines key terms like differential equation, ordinary differential equation, partial differential equation, order, degree, and particular and general solutions. It then provides methods for solving various types of first order differential equations, including separable, homogeneous, exact, linear, and Bernoulli equations. Specific examples are given to illustrate each method.
1. The document provides information on computer aided engineering graphics including notation conventions, quadrant patterns, orthographic projections of points and lines, and projections of planes and 3D objects.
2. Key concepts covered include first angle and third angle projection methods, determining the front, top, and side views of objects, and how to represent inclined lines and planes through reduced views and rotations of reference planes.
3. Examples are given of orthographic projections for points in different quadrants, straight lines with various orientations, planes with different inclinations, and multi-view projections of 3D objects using both first and third angle methods.
This document analyzes correlations between characteristics of different whiskies. It creates a correlation matrix between 12 whisky characteristics for 67 distilleries. It then transforms this into a graph with distilleries as nodes and correlations above 0.8 as edges. Various graph metrics and visualizations are calculated and displayed, including communities detected within the graph.
This document summarizes rules of inference in propositional logic. It defines an argument as a sequence of propositions where all but the final proposition are premises and the final is the conclusion. An argument is valid if the truth of the premises implies the truth of the conclusion. Various rules of inference are provided, including modus ponens, modus tollens, and hypothetical syllogism. Examples are given of identifying and applying different rules of inference to determine the validity of arguments.
Proofs Methods and Strategy
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 10, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
This document analyzes Japanese sake brand data to cluster brands and identify similarities. It first calculates cosine distances between brands based on flavor profiles. It then performs hierarchical clustering on the distances to group brands into 6 clusters. Finally, it creates radar charts of average flavor profiles and word clouds of common tags for each cluster to visualize differences between the groups.
The document discusses the knapsack problem, which involves selecting a subset of items that fit within a knapsack of limited capacity to maximize the total value. There are two versions - the 0-1 knapsack problem where items can only be selected entirely or not at all, and the fractional knapsack problem where items can be partially selected. Solutions include brute force, greedy algorithms, and dynamic programming. Dynamic programming builds up the optimal solution by considering all sub-problems.
This document discusses applications of multiple integrals in various fields such as physics, engineering, economics, computer graphics, and more. Multiple integrals are used to calculate volumes, masses, centers of mass, probabilities, and expected values. They also play a key role in applications such as anti-aliasing in computer graphics, lighting calculations, and determining electric and magnetic fields in electromagnetism. The document provides examples of how multiple integrals are used in these applications.
This document provides an overview of predicate logic, including:
- The basic components of predicate logic like variables, predicates, quantifiers, and propositional functions
- Explanations of the universal and existential quantifiers
- How to negate quantified expressions using De Morgan's laws
- Examples of translating statements between English and predicate logic
This document discusses universal and existential quantification. It defines universal quantification as "for all x in the universe of discourse P(x) is true" and uses the symbol ∀x to represent it. Existential quantification is defined as "there exists an x in the universe of discourse for which P(x) is true" and uses the symbol ∃x to represent it. Examples are given to illustrate the meanings of quantified statements using propositional functions P(x). The document also discusses how to disprove universally quantified statements using counterexamples and the logical equivalences between negating quantified statements.
The document provides an overview of calculus and analytical geometry presented by Group D. It discusses various topics related to integration including the history and development of integration, the definition and purpose of integration, common integration rules and formulas like integration by parts and integration by partial fractions, and real-life applications of integration in fields like engineering, medicine, and physics. Examples are provided to demonstrate how to use integration techniques like integration by parts and integration by partial fractions to evaluate definite integrals.
This document introduces predicates and quantifiers in predicate logic. It defines predicates as functions that take objects and return propositions. Predicates allow reasoning about whole classes of entities. Quantifiers like "for all" (universal quantifier ∀) and "there exists" (existential quantifier ∃) are used to make general statements about predicates over a universe of discourse. Examples demonstrate how predicates and quantifiers can express properties and relationships for objects. Laws of quantifier equivalence are also presented.
This document discusses logical expressions and quantifiers. It provides examples of using quantifiers like ∀ and ∃ to represent statements involving sets, predicates and relationships between variables. It also discusses the meanings of multiple quantifiers, bound and free variables, and how to translate statements into logical expressions.
This document provides an introduction to the Master Theorem, which can be used to determine the asymptotic runtime of recursive algorithms. It presents the three main conditions of the Master Theorem and examples of applying it to solve recurrence relations. It also notes some pitfalls in using the Master Theorem and briefly introduces a fourth condition for cases where the non-recursive term is polylogarithmic rather than polynomial.
Discrete Math Chapter 1 :The Foundations: Logic and ProofsAmr Rashed
The document describes Chapter 1 of a textbook on discrete mathematics and its applications. Chapter 1 covers propositional logic, propositional equivalences, predicates and quantifiers, and nested quantifiers. It defines basic concepts such as propositional variables, logical operators, truth tables, logical equivalence, predicates, quantifiers, and translating between logical expressions and English sentences. Examples are provided to illustrate different logical equivalences and how to negate quantified statements. The chapter introduces key foundations of logic and proofs that are important for discrete mathematics.
First order linear differential equationNofal Umair
1. A differential equation relates an unknown function and its derivatives, and can be ordinary (involving one variable) or partial (involving partial derivatives).
2. Linear differential equations have dependent variables and derivatives that are of degree one, and coefficients that do not depend on the dependent variable.
3. Common methods for solving first-order linear differential equations include separation of variables, homogeneous equations, and exact equations.
This document discusses various methods of mathematical proof, including:
1. Direct proofs, which are used to prove statements of the form "If P then Q" by listing statements from P to Q using axioms and inference rules.
2. Proof by contraposition, which proves "If P then Q" by showing "If not Q then not P".
3. Proof by contradiction, which assumes the negation of what is to be proved and arrives at a contradiction.
The document discusses the Method of Frobenius for solving ordinary differential equations (ODEs) with singular points. It states that the solution for such an ODE is given as an infinite series involving powers of x. To determine the coefficients in the series, one substitutes the series solution into the original ODE, equates coefficients of like powers of x, and obtains the indical equation. Solving this indical equation gives the indicial solution and recurrence relations for the coefficients.
The document is an introduction to ordinary differential equations prepared by Ahmed Haider Ahmed. It defines key terms like differential equation, ordinary differential equation, partial differential equation, order, degree, and particular and general solutions. It then provides methods for solving various types of first order differential equations, including separable, homogeneous, exact, linear, and Bernoulli equations. Specific examples are given to illustrate each method.
1. The document provides information on computer aided engineering graphics including notation conventions, quadrant patterns, orthographic projections of points and lines, and projections of planes and 3D objects.
2. Key concepts covered include first angle and third angle projection methods, determining the front, top, and side views of objects, and how to represent inclined lines and planes through reduced views and rotations of reference planes.
3. Examples are given of orthographic projections for points in different quadrants, straight lines with various orientations, planes with different inclinations, and multi-view projections of 3D objects using both first and third angle methods.
This document analyzes correlations between characteristics of different whiskies. It creates a correlation matrix between 12 whisky characteristics for 67 distilleries. It then transforms this into a graph with distilleries as nodes and correlations above 0.8 as edges. Various graph metrics and visualizations are calculated and displayed, including communities detected within the graph.
This document summarizes rules of inference in propositional logic. It defines an argument as a sequence of propositions where all but the final proposition are premises and the final is the conclusion. An argument is valid if the truth of the premises implies the truth of the conclusion. Various rules of inference are provided, including modus ponens, modus tollens, and hypothetical syllogism. Examples are given of identifying and applying different rules of inference to determine the validity of arguments.
Proofs Methods and Strategy
CMSC 56 | Discrete Mathematical Structure for Computer Science
September 10, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
This document analyzes Japanese sake brand data to cluster brands and identify similarities. It first calculates cosine distances between brands based on flavor profiles. It then performs hierarchical clustering on the distances to group brands into 6 clusters. Finally, it creates radar charts of average flavor profiles and word clouds of common tags for each cluster to visualize differences between the groups.
The document discusses the knapsack problem, which involves selecting a subset of items that fit within a knapsack of limited capacity to maximize the total value. There are two versions - the 0-1 knapsack problem where items can only be selected entirely or not at all, and the fractional knapsack problem where items can be partially selected. Solutions include brute force, greedy algorithms, and dynamic programming. Dynamic programming builds up the optimal solution by considering all sub-problems.
This document discusses applications of multiple integrals in various fields such as physics, engineering, economics, computer graphics, and more. Multiple integrals are used to calculate volumes, masses, centers of mass, probabilities, and expected values. They also play a key role in applications such as anti-aliasing in computer graphics, lighting calculations, and determining electric and magnetic fields in electromagnetism. The document provides examples of how multiple integrals are used in these applications.
This document provides an overview of predicate logic, including:
- The basic components of predicate logic like variables, predicates, quantifiers, and propositional functions
- Explanations of the universal and existential quantifiers
- How to negate quantified expressions using De Morgan's laws
- Examples of translating statements between English and predicate logic
This document discusses universal and existential quantification. It defines universal quantification as "for all x in the universe of discourse P(x) is true" and uses the symbol ∀x to represent it. Existential quantification is defined as "there exists an x in the universe of discourse for which P(x) is true" and uses the symbol ∃x to represent it. Examples are given to illustrate the meanings of quantified statements using propositional functions P(x). The document also discusses how to disprove universally quantified statements using counterexamples and the logical equivalences between negating quantified statements.
Predicate logic uses predicates to describe properties or relations among objects. Predicates are represented by propositional functions like P(x) which denotes "x is a student". Predicates are quantified using quantifiers like the existential quantifier ∃ and universal quantifier ∀. The existential quantifier denotes there exists an object with a certain property, while the universal quantifier denotes a property is true for all objects. When quantified propositions are negated, an existentially quantified proposition becomes universally quantified, and vice versa, according to De Morgan's laws.
The document defines propositional functions as statements involving variables, gives examples, and discusses determining the truth value when values are substituted for variables. It also defines universal and existential quantification using symbols like ∀ and ∃, provides examples, and explains how to negate quantified statements and disprove them with counterexamples.
This document provides an introduction to predicate logic and quantifiers. It begins with terminology like propositional functions, arguments, and universe of discourse. It then defines and provides examples of quantifiers like universal and existential quantifiers. It discusses how to mix quantifiers and their truth values. It also covers binding variables, scope, and negation of quantified statements. Finally, it provides a brief introduction to Prolog, a logic programming language based on predicate logic.
First-order logic (FOL) extends propositional logic by allowing the representation of objects, properties, relations, and functions. It can represent more complex statements than propositional logic. FOL uses constants to represent objects, predicates to represent properties and relations between objects, and quantifiers like "all" and "some" to make generalized statements. Well-formed formulas in FOL contain terms formed from constants and variables, atomic sentences using predicates on terms, and complex sentences combining atomic sentences with logical connectives and quantifiers so that all variables are bound. FOL allows powerful representation of natural language statements about relationships between objects in a domain.
1. The document discusses universal quantification and quantifiers. Universal quantification refers to statements that are true for all variables, while quantifiers are words like "some" or "all" that refer to quantities.
2. It explains that a universally quantified statement is of the form "For all x, P(x) is true" and is defined to be true if P(x) is true for every x, and false if P(x) is false for at least one x.
3. When the universe of discourse can be listed as x1, x2, etc., a universal statement is the same as the conjunction P(x1) and P(x2) and etc., because this
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
The document provides an overview of predicate logic, including:
- Predicates and quantifiers are introduced as the building blocks of predicate logic. Predicates allow representing properties and relations, while quantifiers like "for all" and "there exists" are used to make statements about predicates.
- Examples demonstrate how predicates and quantifiers can be used to represent concepts in logic and translate statements between English and logical expressions.
- Key concepts like universal and existential quantification, propositional functions, logical equivalences for quantifiers, and translating between English and logical expressions are defined and illustrated with examples.
- The document also discusses domains of discourse, precedence of quantifiers, thinking of quantifiers as conjunction
This document discusses predicates and quantifiers in predicate logic. Predicate logic can express statements about objects and their properties, while propositional logic cannot. Predicates assign properties to variables, and quantifiers specify whether a predicate applies to all or some variables in a domain. There are two types of quantifiers: universal quantification with ∀ and existential quantification with ∃. Quantified statements involve predicates, variables ranging over a domain, and quantifiers to specify the scope of the predicate.
The document discusses predicate logic, which extends propositional logic to permit reasoning about classes of entities through the use of predicates and variables. Predicate logic uses predicates that relate variables to form propositions, and quantifiers like "for all" and "there exists" to specify whether predicates apply to all or some members of a universe of discourse. Well-formed formulas in predicate logic contain quantified variables and avoid free variables to form unambiguous propositions.
The document discusses predicate logic, which extends propositional logic to permit reasoning about classes of entities through the use of predicates and variables. Predicate logic uses predicates that relate variables to form propositions, and quantifiers like "for all" and "there exists" to specify whether predicates apply to all or some members of a universe of discourse. Well-formed formulas in predicate logic contain quantified variables and avoid free variables to form unambiguous propositions.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
The axiomatic power of Kolmogorov complexity lbienven
1. The document discusses random axioms and probabilistic proofs in Peano arithmetic. It describes a proof strategy where one could randomly select an integer n that satisfies some formula φ and add it as a new axiom.
2. While this intuition of probabilistic proofs makes sense, it is not really useful since any statement provable with sufficiently high probability is already provable in PA. However, probabilistic proofs can be exponentially more concise than deterministic proofs.
3. The document also discusses Kolmogorov complexity and how statements about it relate to the provability of PA. It can be shown that if C(x) is less than some value, PA will prove it, but PA will never prove a
1) Propositional functions are propositions that contain variables and have no truth value until the variables are assigned values or quantified.
2) Quantifiers like "for all" (universal quantification) and "there exists" (existential quantification) are used to bind variables and give propositional functions truth values.
3) Quantification can be thought of as nested loops over variables, with universal quantification checking for truth at each value and existential checking for at least one true value.
The document describes several common probability distributions used to model random phenomena, including the binomial, geometric, negative binomial, Poisson, uniform, and exponential distributions. It provides the probability mass or density functions that define each distribution, as well as the mean and variance formulas. Examples are given for how each distribution can be applied to problems involving random events like coin flips, dice rolls, polling, customer arrivals, and more.
The document discusses predicate logic and quantifiers. It defines predicate logic as using variables, quantifiers, and predicates to make statements about subjects. It explains common quantifiers like "for all" and "there exists" and how they are used. It also discusses how to translate statements with nested quantifiers and how to determine if quantified statements are true or false.
The document discusses inference rules for quantifiers in discrete mathematics. It provides examples of using universal instantiation, universal generalization, existential instantiation, and existential generalization. It also discusses the rules of universal specification and universal generalization in more detail with examples. Finally, it presents proofs involving quantifiers over integers to demonstrate techniques like direct proof, proof by contradiction, and proving statements' contrapositives.
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Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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1. LECTURE 6
Chapter 1.3
Predicates and Quantifiers
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
2. QUANTIFIER
UNIVERSAL QUANTIFIER
1. Symbol: ∀xP(x)
2. Read as:
“for all 𝒙𝑷(𝒙)"
or
“ for every 𝒙𝑷(𝒙).’’
EXISTENTIAL QUANTIFIER
1. Symbol: ∃𝑥𝑃(𝑥)
2. Read as:
"There is an x such that
𝑷(𝒙) ,"
"There is at least one x such
that 𝑷(𝒙) ,"
or
"For some 𝒙𝑷(𝒙)"
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
3. THE UNIQUENESS QUANTIFIER
Definition:
The uniqueness quantifier of P(x) is the proposition
“There exists a unique x such that P(x) is true.”
The uniqueness quantifier is denoted by ∃! 𝒙𝑷 𝒙 𝒐𝒓 ∃𝟏𝒙𝑷 𝒙
Here ∃! 𝒐𝒓 ∃𝟏 is called the uniqueness quantifier.
The uniqueness quantification ∃! 𝒙𝑷 𝒙 𝒐𝒓 ∃𝟏𝒙𝑷 𝒙 is read as
"There is exactly one x such that 𝑷(𝒙) is true ,"
"There is one and only one x such that 𝑷(𝒙) is true,"
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
4. PRECEDENCE OF QUANTIFIER
The quantifiers ∀ and ∃ have higher precedence then all logical
operators from propositional calculus.
For example,
∀x P(x) Q(x) is the disjunction of ∀x P(x) and Q(x).
In other words, it means (∀x P(x)) Q(x) ) rather than
∀x ( P(x) Q(x))
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
5. Logical Equivalences Involving Quantifiers
Definition:
Statements involving predicates and quantifiers are logically
equivalent if and only if they have the same truth value no
matter which predicates are substituted into these statements and
which domain of discourse is used for the variables in these
propositional functions.
We use the notation S ≡ T to indicate that two statements S and T
involving predicates and quantifiers are logically equivalent.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
6. EXAMPLE 1:
Show that ∀x (P(x) Q(x)) and ∀x P(x) ∀x Q(x) are logically
equivalent (where the same domain is used).
Solution: Suppose we have particular predicates P and Q, with a
common domain. We can show that ∀x (P(x) Q(x)) and ∀x P(x)
∀x Q(x) are logically equivalent by doing two things.
1. First, we show that if ∀x (P(x) Q(x)) is true, then ∀x P(x)
∀x Q(x) is true.
2. Second, we show that if ∀x P(x) ∀x Q(x) is true, then ∀x
(P(x) Q(x)) is true.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
7. Proof of 1st Part:
Suppose that ∀x (P(x) Q(x)) is true. This means that if a is in the
domain, then P(a) Q(a) is true. Hence, P(a) is true and Q(a) is true.
Because P(a) is true and Q(a) is true for every element in the
domain, we can conclude that ∀x P(x) and ∀x Q(x) are both true.
This means that ∀x P(x) ∀x Q(x) is true.
Proof of 2nd Part:
Suppose that ∀x P(x) ∀x Q(x) is true. It follows that ∀x P(x) is
true and ∀x Q(x) is true. Hence, if a is in the domain, then P(a) is
true and Q(a) is true [because P(x) and Q(x) are both true for all
elements in the domain, there is no conflict using the same value of a
here].
It follows that for all a, P(a) Q(a) is true. It follows that ∀x (P(x)
Q(x)) is true. We can now conclude that
∀x (P(x) Q(x)) ≡ ∀x P(x) ∀x Q(x)
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
8. De Morgan's Laws for Quantifiers
1. ¬∀x P(x) ≡ ∃𝐱¬ P(x).
2. ¬∃ xP(x) ≡ ∀x¬ P(x).
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
9. Negating Quantified Expressions
Example 1: Consider the statement:
"Every student in your class has taken a course in calculus.” This statement is
a universal quantification, namely, ∀x P(x) .
∀x P(x)= "Every student in your class has taken a course in calculus.” The
negation of this statement is
¬∀x P(x) ="It is not the case that every student in your class has taken a
course in calculus.” …..(1)
Where P(x) = "x has taken a course in calculus" and the domain consists of the
students in your class.
¬ P(x) = "x has not taken a course in calculus“
Above statement (1) equivalent to
"There is a student in your class who has not taken a course in calculus.”
And this is simply the existential quantification of the negation of the original
propositional function, namely, ∃𝐱¬ P(x).
∃𝐱¬ P(x)= "There is a student in your class who has not taken a course in
calculus.”
That is ¬∀x P(x) ≡ ∃𝐱¬ P(x).
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
10. ¬∀x P(x) ≡ ∃𝐱¬ P(x).
It is not the case that every student in
your class has taken a course in
calculus.
There is a student in your class who has
not taken a course in calculus
ব্যাপারটা এমন নয় যে আপনার ক্লাসের
প্রসযযক ছাত্রই কযালক
ু লাে যকাে সকসরসছ
আপনার ক্লাসে একজন ছাত্র আসছ যে
কযালক
ু লাসের যকাে সকসরনন
¬∃ xP(x) ≡ ∀x¬ P(x).
It is not the case that there is a student
in this class who has taken a course in
calculus.
Every student in this class has not taken
calculus.
ব্যাপারটা এমন নয় যে এই ক্লাসে একজন
ছাত্র আসছ যে কযালক
ু লাে যকাে সকসরসছ।
এই ক্লাসের প্রসযযক নিক্ষার্থী কযালক
ু লাে
যনয়নন।
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
11. Example 2: Consider the statement:
" There is a student in this class who has taken a course in calculus.” This
statement is a existential quantification, namely, ∃ x P(x) .
∃ x P(x) = " There is a student in this class who has taken a course in
calculus.”
The negation of this statement is
¬∃xP(x)=“It is not the case that there is a student in this class who has taken
a course in calculus."…..(1)
Where P(x) = "x has taken a course in calculus" and the domain consists of the
students in your class.
¬ P(x) = "x has not taken a course in calculus“
Above statement (1) equivalent to
"Every student in this class has not taken calculus.”
And this is simply the universal quantification of the negation of the original
propositional function, namely, ∀x¬ P(x).
∀x¬ P(x)= "Every student in this class has not taken calculus”
That is ¬∃ xP(x) ≡ ∀x¬ P(x).
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
12. ¬∀x P(x) ≡ ∃𝐱¬ P(x).
It is not the case that every student in
your class has taken a course in
calculus.
There is a student in your class who has
not taken a course in calculus
ব্যাপারটা এমন নয় যে আপনার ক্লাসের
প্রসযযক ছাত্রই কযালক
ু লাে যকাে সকসরসছ
আপনার ক্লাসে একজন ছাত্র আসছ যে
কযালক
ু লাসের যকাে সকসরনন
¬∃ xP(x) ≡ ∀x¬ P(x).
It is not the case that there is a student
in this class who has taken a course in
calculus.
Every student in this class has not taken
calculus.
ব্যাপারটা এমন নয় যে এই ক্লাসে একজন
ছাত্র আসছ যে কযালক
ু লাে যকাে সকসরসছ।
এই ক্লাসের প্রসযযক নিক্ষার্থী কযালক
ু লাে
যনয়নন।
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
13. Example 3: What are the negations of the statements "There is an honest
politician"
Solution: Consider the statement: “There is an honest politician.” This statement
is a existential quantification, namely, ∃ x P(x) .
∃ xP(x) = " There is an honest politician.”
The negation of this statement is
¬∃ 𝒙𝑷(𝒙) = “It is not the case that there is an honest politician."…..(1)
Where P(x) = “x is honest" and the domain consists of all politicians.
¬ P(x) = "x is dishonest“
Above statement (1) equivalent to
"Every politician is dishonest."
And this is simply the universal quantification of the negation of the original
propositional function, namely, ∀x¬ P(x).
∀x¬ P(x)= "Every politician is dishonest”
That is ¬∃ xP(x) ≡ ∀x¬ P(x).
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
14. Example 4: What are the negations of the statements " All Americans eat
cheeseburgers " .
Solution: Consider the statement: "All Americans eat cheeseburgers.” This
statement is a universal quantification, namely, ∀x P(x) .
∀x P(x)= " All Americans eat cheeseburgers.”
The negation of this statement is
¬∀x P(x) ="It is not the case that All Americans eat cheeseburgers.” …..(1)
Where P(x) = “x eats cheeseburgers” and the domain consists of the students in
your class.
¬ P(x) = " x does not eat cheeseburgers“
Above statement (1) equivalent to
"There is an American who does not eat cheeseburgers.”
And this is simply the existential quantification of the negation of the original
propositional function, namely, ∃𝐱¬ P(x).
∃𝐱¬ P(x)= "There is an American who does not eat cheeseburgers.”
That is ¬∀x P(x) ≡ ∃𝐱¬ P(x).
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
15. Example 5: What are the negations of the statements:
i. ∀x (𝒙𝟐
> 𝒙)
ii. ∃x 𝒙𝟐 = 𝟐
Solution: De Morgan's Laws for Quantifiers
1. ¬∀x P(x) ≡ ∃𝐱¬ P(x).
2. ¬∃ xP(x) ≡ ∀x¬ P(x).
i. Applying De Morgan's 1st Law:¬∀x P(x) ≡ ∃𝐱¬ P(x).
¬∀x (𝒙𝟐> 𝒙) ≡ ∃𝐱¬ 𝒙𝟐 > 𝒙
¬∀x (𝒙𝟐> 𝒙) ≡ ∃𝐱 𝒙𝟐 ≤ 𝒙
ii. Applying De Morgan's 2nd Law:¬∃ xP(x) ≡ ∀x¬ P(x).
¬∃ x 𝒙𝟐 = 𝟐 ≡ ∀x¬ 𝒙𝟐 = 𝟐
¬∃ x 𝒙𝟐
= 𝟐 ≡ ∀x 𝒙𝟐
≠ 𝟐
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
16. Example 6:
Show that, ¬∀x (P(x)→ 𝐐(𝐱)) ≡ ∃𝐱 (P(x)¬𝑸(𝒙)).
Solution: According to De Morgan's 1st Law:
¬∀x P(x) ≡ ∃𝐱¬ P(x).
¬∀x (P(x)→ 𝐐(𝐱)) ≡ ∃𝐱¬ (P(x)→ 𝐐(𝐱))
≡ ∃𝐱 (P(x)¬𝐐(𝐱)) [¬(𝐴 → 𝐵) = 𝐴¬𝐵)]
[Proved]
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
17. Problem 1: Let C(x) be the statement “x has a cat”, let D(x) be the statement
“x has a dog”, and let F(x) be the statement “x has a ferret”. Express each of
these statements in terms of C(x), D(x), F(x), quantifiers, and logical
connectives. Let the domain consist of all students in your class.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
Question Solution
a. A student in your class has a cat, a dog, and a
ferret.
∃𝐱 (𝐂(x)D(x) 𝑭(𝒙)).
b. All students in your class have a cat, a dog, or a
ferret.
∀𝐱 (𝐂(x)D(x)𝐅(𝒙)).
c. Some student in your class has a cat and a ferret,
but not a dog
∃𝐱 (𝐂(x)¬D(x) 𝑭(𝒙)).
d. No student in your class has a cat, a dog, and a
ferret.
¬∃𝐱 (𝐂(x)D(x) 𝑭(𝒙)).
e. For each of the three animals, cats, dogs, and
ferrets, there is a student in your class who has
one of these animals as a pet
∃𝐱 (𝐂(x)D(x) 𝑭(𝒙)).
18. Problem 2: Let Q(x) be the statement “𝑥 + 1 > 2𝑥”. If the domain consists of
all integers, what are the truth values?
a) Q(0) b) Q(−1) c) Q(1) d) ∃x Q(x)
e) ∀x Q(x) f) ∃x ¬Q(x) g) ∀x ¬Q(x)
Solution:
a) Since 0 + 1 > 0 · we know that Q(0) is true.
b) Since (−1) + 1 > 2 (−1) · we know that Q(−1) is true.
c) Since 1 + 1 > 2 · we know that Q(1) is false.
d) From part (a) we know that there is at least one x that makes Q(x) true, so
∃x Q(x) is true.
e) From part (c) we know that there is at least one x that makes Q(x) false, so
∀x Q(x) is false.
f) From part (c) we know that there is at least one x that makes Q(x) false, so
∃x ¬Q(x) is true.
g) From part (a) we know that there is at least one x that makes Q(x) true, so
∀x ¬Q(x) is false.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
20. The Order of Quantifiers
Example 1: Let P (x , y) be the statement "x + y = y + x ." What are the truth values of
the quantifications ∀x ∀y P(x , y) and ∀y ∀x P (x , y) where the domain for all
variables consists of all real numbers?
Solution: The quantification ∀x ∀y P(x , y) denotes the proposition
"For all real numbers x , for all real numbers y, x + y = y + x .“
Because P(x, y) is true for all real numbers x and y, then , the proposition ∀x ∀y P(x ,
y) is true.
The quantification ∀y ∀x P(x , y) denotes the proposition
"For all real numbers y , for all real numbers x, x + y = y + x .“
Because P(x, y) is true for all real numbers x and y, then , the proposition ∀y ∀x P(x ,
y) is true.
That is, ∀x ∀y P(x , y) and ∀y ∀x P(x , y) have the same meaning, and both are true.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
21. Example 2: Let Q (x , y) be the statement "x + y = 0." What are the truth
values of the quantifications ∃y∀x Q(x , y) and ∀x∃y Q (x , y) where the
domain for all variables consists of all real numbers?
Solution: The quantification ∃y∀x Q(x , y) denotes the proposition
"There is a real number y such that for every real number x, Q(x , y).”
No matter what value of y is chosen, there is only one value of x for which x +
y = 0. Because there is no real number y such that x + y = 0 for all real
numbers x , the statement ∃y∀x Q(x , y) is false.
The quantification ∀x ∃y Q (x , y) denotes the proposition
"For every real number x, there is a real number y such that Q(x , y).
Given a real number x , there is a real number y such that x + y = 0; namely,
y= -x . Hence, the statement ∀x ∃y Q (x , y) is true. The statements ∃y∀x Q(x ,
y) and ∀x∃y Q (x , y) are not logically equivalent.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
22. Example 3: Let Q (x , y, z) be the statement "x + y = z." What are the truth
values of the quantifications ∀x ∀y ∃z Q(x , y, z) and ∃y ∀x ∀y Q (x , y, z)
where the domain for all variables consists of all real numbers?
Solution: The quantification ∀x ∀y ∃z Q(x , y, z) denotes the proposition
"For all real numbers x and for all real numbers y there is a real
number z such that x + y = z.”
Suppose that x and y are assigned values. Then, there exists a real number z
such that x + y = z. then the quantification ∀x ∀y ∃z Q(x , y, z) is true.
The quantification ∃y ∀x ∀y Q(x, y, z) denotes the proposition
“There is a real number z such that for all real numbers x and for all
real numbers y it is true that x + y = z”
The quantification ∃y ∀x ∀y Q(x, y, z) is false, because there is no value of z
that satisfies the equation x + y = z for all values of x and y.
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
23. Assignment 4
1. Let P(x) be the statement "x spends more than five hours every weekday in
class," where the domain for x consists of all students. Express each of
these quantifications in English.
i. ∃ x P(x) ii. ∀x P(x) iii. ∀x¬ P(x) iv. ∃𝐱¬ P(x).
2. Let P(x) be the statement "x can speak Russian" and let Q(x) be the
statement "x knows the computer language C++." Express each of these
sentences in terms of P(x), Q(x), quantifiers, and logical connectives. The
domain for quantifiers consists of all students at your school.
a) There is a student at your school who can speak Russian and who knows
C++.
b) There is a student at your school who can speak Russian but who doesn't
know C++.
c) Every student at your school either can speak Russian or knows C++.
d) No student at your school can speak Russian or knows C++
Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
24. Prepared by Khairun Nahar,Assistant Professor,
Department of CSE, Comilla University
3.