This document summarizes key concepts about predicates and quantifiers. It defines universal and existential quantifiers, as well as the uniqueness quantifier. It discusses precedence of quantifiers and logical equivalences involving quantifiers. Examples are provided to illustrate negating quantified expressions using De Morgan's laws for quantifiers. The document also contains practice problems applying these concepts.
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.
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.
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 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 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.
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.
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.
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 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 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 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.
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.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
This document discusses quantifiers and open statements. It defines universal and existential quantifiers and provides examples of open statements involving variables. Several quantified statements are expressed symbolically and evaluated for truth value. Universal statements about integers being perfect squares, positive, even, or divisible by 3 or 7 are determined to be true or false.
This document discusses predicates and quantifiers in predicate logic. It begins by explaining the limitations of propositional logic in expressing statements involving variables and relationships between objects. It then introduces predicates as statements involving variables, and quantifiers like universal ("for all") and existential ("there exists") to express the extent to which a predicate is true. Examples are provided to demonstrate how predicates and quantifiers can be used to represent statements and enable logical reasoning. The document also covers translating statements between natural language and predicate logic, and negating quantified statements.
This document presents a new model of decision making under risk and uncertainty called the Harmonic Probability Weighting Function (HPWF) model. The HPWF model incorporates mental states using a weak harmonic transitivity axiom and an abstract harmonic representation of noise. It explains phenomena like the conjunction fallacy and preference reversal. The HPWF uses a harmonic component controlled by a phase function to characterize how a decision maker's mental states influence probability weighting. Maximum entropy methods can be used to derive a coherent harmonic probability weighting function from the HPWF model.
This document provides an overview of one-dimensional random variables including definitions, types (discrete vs continuous), and probability distributions. It defines a random variable as a function that assigns a numerical value to each outcome of a random experiment. Random variables can be either discrete, taking on countable values, or continuous, assuming any value in an interval. The probability distribution of a discrete random variable is defined by a probability mass function, while a continuous random variable has a probability density function. Examples are given of both types of random variables and their distributions.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
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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.
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Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
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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).
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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]
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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.