This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
Here is a proof of this statement using resolution refutation:
1. ∀x∀y(F(x) ∧ F(y) ∧ L(x,y)) → S(x,y) (Premise: Any fish larger can swim faster)
2. ∃x∀y(F(y) → L(x,y)) (Premise: There exists a largest fish)
3. ∃x∀y(F(y) → S(x,y)) (Goal: There exists a fastest fish)
4. F(a) ∧ F(b) ∧ L(a,b) → S(a
The document defines binary relations and provides examples of binary relations between sets. It then discusses properties of binary relations such as being reflexive, symmetric, transitive, complete, antisymmetric, asymmetric, or irreflexive. It introduces the concepts of preorders, orders, equivalence relations, and partitions. A preorder is a binary relation that is transitive and either reflexive or irreflexive. An order is a complete, transitive, and antisymmetric preorder. An equivalence relation is a reflexive, symmetric, and transitive binary relation that partitions a set into equivalence classes. Utility functions are introduced as a way to represent preorders, where a utility function u represents a preorder R if xRy if and
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
This document provides an overview of affine algebraic groups and group actions on algebraic varieties. Some key points:
1. An affine algebraic group G is an affine algebraic variety that is also a group, such that multiplication and inverse maps are morphisms of varieties. Examples include GLn, SLn, finite groups.
2. A group G acts on a variety X if the map G × X to X given by the action is a morphism. Orbits are open in their closure. There is always a closed orbit.
3. The connected component G° of the identity in G is a closed normal subgroup of finite index, and any closed subgroup of finite index contains G°.
This document provides an introduction to formal logic and mathematical logic. It defines key concepts like logic, propositions, statements, truth values, and logical connectives. It also differentiates between propositions and non-propositions, and statements and non-statements. Examples are given of propositions and non-propositions. The basics of propositional logic and how to determine the truth values of expressions are explained.
We prove that the infinite composition of continuous surjective mappings defined in compact set is a surjective mapping if the infinite composition is a well-defined mapping.
This document introduces equivalence relations and partitions. It defines an equivalence relation as a binary relation that is reflexive, symmetric, and transitive. Equivalence relations partition a set into disjoint equivalence classes that cover the entire set. The quotient set of a set by an equivalence relation consists of the equivalence classes. Every equivalence relation determines a partition, and every partition determines an equivalence relation. Examples are provided to illustrate these concepts using the equivalence relation of congruence modulo 3 on the integers.
Here is a proof of this statement using resolution refutation:
1. ∀x∀y(F(x) ∧ F(y) ∧ L(x,y)) → S(x,y) (Premise: Any fish larger can swim faster)
2. ∃x∀y(F(y) → L(x,y)) (Premise: There exists a largest fish)
3. ∃x∀y(F(y) → S(x,y)) (Goal: There exists a fastest fish)
4. F(a) ∧ F(b) ∧ L(a,b) → S(a
The document defines binary relations and provides examples of binary relations between sets. It then discusses properties of binary relations such as being reflexive, symmetric, transitive, complete, antisymmetric, asymmetric, or irreflexive. It introduces the concepts of preorders, orders, equivalence relations, and partitions. A preorder is a binary relation that is transitive and either reflexive or irreflexive. An order is a complete, transitive, and antisymmetric preorder. An equivalence relation is a reflexive, symmetric, and transitive binary relation that partitions a set into equivalence classes. Utility functions are introduced as a way to represent preorders, where a utility function u represents a preorder R if xRy if and
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than propositional logic.
The topics on first order logic are:
1-Quantifiers
2-Negation
3-Multiple quantifiers
4-Arguments of quantified statements
This document provides an overview of affine algebraic groups and group actions on algebraic varieties. Some key points:
1. An affine algebraic group G is an affine algebraic variety that is also a group, such that multiplication and inverse maps are morphisms of varieties. Examples include GLn, SLn, finite groups.
2. A group G acts on a variety X if the map G × X to X given by the action is a morphism. Orbits are open in their closure. There is always a closed orbit.
3. The connected component G° of the identity in G is a closed normal subgroup of finite index, and any closed subgroup of finite index contains G°.
This document provides an introduction to formal logic and mathematical logic. It defines key concepts like logic, propositions, statements, truth values, and logical connectives. It also differentiates between propositions and non-propositions, and statements and non-statements. Examples are given of propositions and non-propositions. The basics of propositional logic and how to determine the truth values of expressions are explained.
We prove that the infinite composition of continuous surjective mappings defined in compact set is a surjective mapping if the infinite composition is a well-defined mapping.
This document provides an introduction to propositional logic and first-order logic. It defines propositional logic, including propositional variables, connectives like conjunction and disjunction, and the laws of propositional logic. It then introduces first-order logic, which adds quantifiers, variables, functions, and predicates to represent objects, properties, and relations in a domain. First-order logic allows for more expressive statements about individuals and generalizations than propositional logic alone.
This document discusses predicates and quantifiers in predicate logic. It defines predicates as statements involving variables whose truth depends on variable values. Predicates become propositions when variables are assigned values. Quantifiers like universal (∀) and existential (∃) are used to express the extent to which a predicate is true. Universal quantifiers mean a predicate is true for all variables, while existential quantifiers mean a predicate is true for at least one variable. Examples show how to represent English language sentences using predicates and quantifiers in predicate logic.
Minimality in homological PDE- Charles E. Rodgers - 2019TimWiseman12
Abstract. Let Z be a covariant monoid. In [16], it is shown that there exists a real and composite set. We show that there exists a Liouville,
Fermat and composite super-meromorphic, super-bijective, quasi-Smale probability space. O. White [16] improved upon the results of J. Taylor by computing Pythagoras, pairwise anti-compact, freely linear numbers.
Therefore a central problem in hyperbolic model theory is the computation of linearly open hulls.
This document provides an introduction to logic, including propositional logic and predicate calculus. It defines key concepts such as logical values, propositions, operators, truth tables, logical expressions, worlds, models, inference rules, quantification, and definitions. Propositional logic manipulates true and false values using operators like AND and OR. Predicate calculus extends this to allow predicates, constants, functions, and quantification over variables. Inference involves applying rules to derive new statements, but the search space grows too large to feasibly perform by hand.
This document provides an overview of the topics that will be covered in a finite mathematics course, including residue arithmetic, elements of finite groups/rings/fields, number theory concepts like the Euclidean algorithm and Chinese Remainder Theorem, and basics of finite vector spaces and fields. The style of the course will be leisurely and discursive, focusing on mathematical thinking and discovery. While the mathematics is classical, it will be new to students. The goal is to emphasize elegance and aesthetics over utility alone.
This paper introduces the concept of h-integrability as a condition for obtaining weak laws of large numbers for arrays (WLLNFA). The paper defines various types of uniform integrability including Cesaro uniform integrability, -uniform integrability, Cesaro α-integrability, and h-integrability. It is shown that h-integrability is a weaker condition than these other forms of uniform integrability. The main results of the paper establish that if an array is h-integrable with exponent r, then the sample mean of the array converges in probability to the expected value under certain conditions. This extends previous results on obtaining weak laws of large numbers for dependent random variables.
The document discusses propositional logic and covers topics like propositional variables, truth tables, logical equivalence, predicates, and quantifiers. It defines key concepts such as propositions, tautologies, contradictions, predicates, universal and existential quantifiers. Examples are provided to illustrate different types of truth tables, logical equivalences like De Morgan's laws, and uses of quantifiers.
This document discusses linear, abelian, and continuous groups and how relaxing these properties leads to more complex groups. It begins with the simplest group, the real numbers R, and progresses to integer lattices Z and Z^n, then non-abelian Lie groups like SL(n,R). Lattices in these groups like SL(n,Z) are discussed, along with properties like the congruence subgroup property. Open questions are raised regarding the irreducibility of random matrices and deciding membership in subgroups of SL(n,Z).
This document discusses different types of proofs in mathematics including direct proofs, existence proofs, and disproving statements with counterexamples. It provides examples of direct proofs showing that the sum of two even integers is even and that the square of an odd integer is odd. It also gives an example of an existence proof showing there is an even number that can be written as the sum of two prime numbers. Finally, it discusses how to disprove universal statements by finding a counterexample.
This document summarizes key concepts in topology related to the cluster point or limit point of a set in a topological space. It defines a cluster/limit point as a point such that every neighborhood of it intersects the set in at least one other point. Theorems establish that a set and its cluster points together form its closure, and that a set is closed if it contains all its cluster points. No finite set can have a cluster point. The union of a set's cluster points with the set of another's forms the cluster points of their union.
The document defines and provides examples of rings and ideals. Some key points:
- A ring consists of a set with operations of addition and multiplication satisfying certain properties like commutativity and associativity.
- Common examples of rings include the integers, rational numbers, real numbers, polynomials, and matrices.
- An ideal is a subset of a ring that is closed under addition and multiplication. Ideals play an important role in ring theory.
- Given an ideal I of a ring R, a quotient ring R/I can be constructed by identifying elements of R that differ by an element of I. Operations are defined on these equivalence classes.
Devaney Chaos Induced by Turbulent and Erratic FunctionsIOSRJM
Let I be a compact interval and f be a continuous function defined from I to I. We study the relationship between tubulent function, erratic function and Devaney Chaos.
The document presents information on fuzzy equivalence relations. It defines fuzzy equivalence relations as fuzzy relations that are reflexive, symmetric, and transitive. Examples of fuzzy relations that satisfy these properties are provided. Related theorems state that a fuzzy relation is symmetric if its alpha cuts are symmetric relations, and transitive if its alpha cuts are transitive relations. Sample problems involving properties of compositions of fuzzy relations are outlined. The presentation concludes with a bibliography reference.
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 famous Kruskal's tree theorem states that the collection of finite trees labelled over a well quasi order and ordered by homeomorphic embedding, forms a well quasi order. Its intended mathematical meaning is that the collection of finite, connected and acyclic graphs labelled over a well quasi order is a well quasi order when it is ordered by the graph minor relation.
Oppositely, the standard proof(s) shows the property to hold for trees in the Computer Science's sense together with an ad-hoc, inductive notion of embedding. The mathematical result follows as a consequence in a somewhat unsatisfactory way.
In this talk, a variant of the standard proof will be illustrated explaining how the Computer Science and the graph-theoretical statements are strictly coupled, thus explaining why the double statement is justified and necessary.
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.
The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.
The document is a presentation about using model theory to prove Hilbert's Weak Nullstellensatz. It begins with introductions to model theory, including definitions of structures, embeddings, elementary extensions, theories, and model-completeness. It then states that the theory of algebraically closed fields has model-completeness. The presentation concludes with a proof of the Weak Nullstellensatz using these model-theoretic concepts, showing there is a tuple in an algebraically closed field that satisfies a given ideal of polynomials.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)Jan Plaza
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
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 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 provides an introduction to propositional logic and first-order logic. It defines propositional logic, including propositional variables, connectives like conjunction and disjunction, and the laws of propositional logic. It then introduces first-order logic, which adds quantifiers, variables, functions, and predicates to represent objects, properties, and relations in a domain. First-order logic allows for more expressive statements about individuals and generalizations than propositional logic alone.
This document discusses predicates and quantifiers in predicate logic. It defines predicates as statements involving variables whose truth depends on variable values. Predicates become propositions when variables are assigned values. Quantifiers like universal (∀) and existential (∃) are used to express the extent to which a predicate is true. Universal quantifiers mean a predicate is true for all variables, while existential quantifiers mean a predicate is true for at least one variable. Examples show how to represent English language sentences using predicates and quantifiers in predicate logic.
Minimality in homological PDE- Charles E. Rodgers - 2019TimWiseman12
Abstract. Let Z be a covariant monoid. In [16], it is shown that there exists a real and composite set. We show that there exists a Liouville,
Fermat and composite super-meromorphic, super-bijective, quasi-Smale probability space. O. White [16] improved upon the results of J. Taylor by computing Pythagoras, pairwise anti-compact, freely linear numbers.
Therefore a central problem in hyperbolic model theory is the computation of linearly open hulls.
This document provides an introduction to logic, including propositional logic and predicate calculus. It defines key concepts such as logical values, propositions, operators, truth tables, logical expressions, worlds, models, inference rules, quantification, and definitions. Propositional logic manipulates true and false values using operators like AND and OR. Predicate calculus extends this to allow predicates, constants, functions, and quantification over variables. Inference involves applying rules to derive new statements, but the search space grows too large to feasibly perform by hand.
This document provides an overview of the topics that will be covered in a finite mathematics course, including residue arithmetic, elements of finite groups/rings/fields, number theory concepts like the Euclidean algorithm and Chinese Remainder Theorem, and basics of finite vector spaces and fields. The style of the course will be leisurely and discursive, focusing on mathematical thinking and discovery. While the mathematics is classical, it will be new to students. The goal is to emphasize elegance and aesthetics over utility alone.
This paper introduces the concept of h-integrability as a condition for obtaining weak laws of large numbers for arrays (WLLNFA). The paper defines various types of uniform integrability including Cesaro uniform integrability, -uniform integrability, Cesaro α-integrability, and h-integrability. It is shown that h-integrability is a weaker condition than these other forms of uniform integrability. The main results of the paper establish that if an array is h-integrable with exponent r, then the sample mean of the array converges in probability to the expected value under certain conditions. This extends previous results on obtaining weak laws of large numbers for dependent random variables.
The document discusses propositional logic and covers topics like propositional variables, truth tables, logical equivalence, predicates, and quantifiers. It defines key concepts such as propositions, tautologies, contradictions, predicates, universal and existential quantifiers. Examples are provided to illustrate different types of truth tables, logical equivalences like De Morgan's laws, and uses of quantifiers.
This document discusses linear, abelian, and continuous groups and how relaxing these properties leads to more complex groups. It begins with the simplest group, the real numbers R, and progresses to integer lattices Z and Z^n, then non-abelian Lie groups like SL(n,R). Lattices in these groups like SL(n,Z) are discussed, along with properties like the congruence subgroup property. Open questions are raised regarding the irreducibility of random matrices and deciding membership in subgroups of SL(n,Z).
This document discusses different types of proofs in mathematics including direct proofs, existence proofs, and disproving statements with counterexamples. It provides examples of direct proofs showing that the sum of two even integers is even and that the square of an odd integer is odd. It also gives an example of an existence proof showing there is an even number that can be written as the sum of two prime numbers. Finally, it discusses how to disprove universal statements by finding a counterexample.
This document summarizes key concepts in topology related to the cluster point or limit point of a set in a topological space. It defines a cluster/limit point as a point such that every neighborhood of it intersects the set in at least one other point. Theorems establish that a set and its cluster points together form its closure, and that a set is closed if it contains all its cluster points. No finite set can have a cluster point. The union of a set's cluster points with the set of another's forms the cluster points of their union.
The document defines and provides examples of rings and ideals. Some key points:
- A ring consists of a set with operations of addition and multiplication satisfying certain properties like commutativity and associativity.
- Common examples of rings include the integers, rational numbers, real numbers, polynomials, and matrices.
- An ideal is a subset of a ring that is closed under addition and multiplication. Ideals play an important role in ring theory.
- Given an ideal I of a ring R, a quotient ring R/I can be constructed by identifying elements of R that differ by an element of I. Operations are defined on these equivalence classes.
Devaney Chaos Induced by Turbulent and Erratic FunctionsIOSRJM
Let I be a compact interval and f be a continuous function defined from I to I. We study the relationship between tubulent function, erratic function and Devaney Chaos.
The document presents information on fuzzy equivalence relations. It defines fuzzy equivalence relations as fuzzy relations that are reflexive, symmetric, and transitive. Examples of fuzzy relations that satisfy these properties are provided. Related theorems state that a fuzzy relation is symmetric if its alpha cuts are symmetric relations, and transitive if its alpha cuts are transitive relations. Sample problems involving properties of compositions of fuzzy relations are outlined. The presentation concludes with a bibliography reference.
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 famous Kruskal's tree theorem states that the collection of finite trees labelled over a well quasi order and ordered by homeomorphic embedding, forms a well quasi order. Its intended mathematical meaning is that the collection of finite, connected and acyclic graphs labelled over a well quasi order is a well quasi order when it is ordered by the graph minor relation.
Oppositely, the standard proof(s) shows the property to hold for trees in the Computer Science's sense together with an ad-hoc, inductive notion of embedding. The mathematical result follows as a consequence in a somewhat unsatisfactory way.
In this talk, a variant of the standard proof will be illustrated explaining how the Computer Science and the graph-theoretical statements are strictly coupled, thus explaining why the double statement is justified and necessary.
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.
The document discusses the pigeonhole principle, countability, and cardinality. The pigeonhole principle states that if n items are placed into m pigeonholes with n > m, then at least one pigeonhole must contain more than one item. Countability refers to sets being either finite or denumerable (having the same cardinality as the natural numbers). Cardinality compares the sizes of sets based on whether a bijection exists between them. The document provides examples and proofs of these concepts.
The document is a presentation about using model theory to prove Hilbert's Weak Nullstellensatz. It begins with introductions to model theory, including definitions of structures, embeddings, elementary extensions, theories, and model-completeness. It then states that the theory of algebraically closed fields has model-completeness. The presentation concludes with a proof of the Weak Nullstellensatz using these model-theoretic concepts, showing there is a tuple in an algebraically closed field that satisfies a given ideal of polynomials.
6.2 Reflexivity, symmetry and transitivity (dynamic slides)Jan Plaza
This document introduces basic concepts of set theory including definitions of reflexive, symmetric and transitive relations. It provides examples of relations between lines that are parallel or perpendicular. It also discusses relations between people like siblings and ancestors. The document proves properties of congruence relations and discusses counting relations with certain properties over sets with a given number of elements. It addresses a claim about relations that are symmetric and transitive necessarily being reflexive, finding a counterexample.
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 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
A Unifying theory for blockchain and AILonghow Lam
This document proposes a unifying theory connecting blockchain and artificial intelligence technologies. It introduces the Lam-Visser theory and how it fits within the Damhof Quadrants framework. The document provides definitions related to the main result, which states that there exists a minimal, ultra-connected, almost everywhere linear and generic solvable, semi-countable polytope if a certain condition is met. It then discusses applications of this theory to questions of associativity and the computation of analytically independent subalgebras.
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.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document contains:
1. Comments on a homework assignment about microeconomics and proofs, noting common mistakes and suggesting a standard proof structure.
2. The homework problems and solutions, with proofs showing preferences are complete, transitive, and representable by a utility function.
3. An appendix theorem about partitioning a set into equivalence classes using a binary relation.
The document summarizes key concepts from a lecture on discrete structures including:
1) Predicates are statements with variables that become true or false when values are substituted. The truth set of a predicate contains values that make the statement true.
2) Universal statements are true if the predicate is true for all values, while existential statements are true if the predicate is true for at least one value.
3) Statements can be translated between formal logic notation using quantifiers and informal English. Negations of universal statements are existential, and vice versa.
The document summarizes key concepts from a lecture on discrete structures, including:
1) It defines predicates as sentences containing variables that become statements when values are substituted, and introduces truth sets as the set of elements making a predicate true.
2) It discusses universal and existential statements, where a universal statement is true if a predicate is true for all variables, and an existential is true if true for at least one variable.
3) It explains translating between formal quantified statements and informal English statements, and shows several examples of translating in both directions.
This document presents an overview of Rossella Marrano's talk on a qualitative perspective on vagueness and degrees of truth. The talk explores representing vagueness through comparative judgments of truth between sentences, rather than precise numerical assignments. It proposes axioms for a binary relation between sentences based on one being "more true" than the other. Representation results show that real-valued truth functions can arise from satisfying the axioms. The approach aims to address objections that assigning precise numerical truth values replaces vagueness with artificial precision.
Contact geometry is the study of certain geometric structures on odd dimensional smooth manifolds. A contact structure is a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability. The associated one form is called a contact form and uniquely determines a Hamiltonian-like vector field called the Reeb vector field on the manifold. I will give some background on this subject, including motivation from classical mechanics. I will also explain how to make use of J-holomorphic curves to obtain a Floer theoretic contact invariant, contact homology, whose chain complex is generated by closed Reeb orbits. This talk will feature numerous graphics to acclimate people to the realm of contact geometry.
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 convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to statistics.
The document provides instruction on linear functions and how to find the x-intercept and y-intercept of a linear equation. It discusses writing linear equations in function notation as f(x), defines the x-intercept as where the graph crosses the x-axis (y = 0) and the y-intercept as where it crosses the y-axis (x = 0). It gives the formula for linear equations in standard form and shows how to set x = 0 and y = 0 to find the intercepts. Examples are worked out, showing how to graph the line based on the intercepts. Classwork assignments are noted at the end.
I am Humphrey J. I am a Math Assignment Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from Las Vegas, USA. I have been helping students with their assignments for the past 11 years. I solved assignments related to Math.
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2. If you have a nonlinear relationship between an independent varia.pdfsuresh640714
2. If you have a nonlinear relationship between an independent variable x and a dependent
variable y, how can you apply the least -squares fit method? Does this work for all nonlinear
relationships? if not please give an example of a nonlinear equation where the fit method cannot
be used.(No need to use Matlab on this question).
Solution
A)The assumptions of linearity and additivity are both implicit in this specification. • Additivity
= assumption that for each IV X, the amount of change in E(Y) associated with a unit increase in
X (holding all other variables constant) is the same regardless of the values of the other IVs in
the model. That is, the effect of X1 does not depend on X2; increasing X1 from 10 to 11 will
have the same effect regardless of whether X2 = 0 or X2 = 1. • With non-additivity, the effect of
X on Y depends on the value of a third variable, e.g. gender. As we’ve just discussed, we use
models with multiplicative interaction effects when relationships are non-additive.
Linearity = assumption that for each IV, the amount of change in the mean value of Y associated
with a unit increase in the IV, holding all other variables constant, is the same regardless of the
level of X, e.g. increasing X from 10 to 11 will produce the same amount of increase in E(Y) as
increasing X from 20 to 21. Put another way, the effect of a 1 unit increase in X does not depend
on the value of X. • With nonlinearity, the effect of X on Y depends on the value of X; in effect,
X somehow interacts with itself. This is sometimes refered to as a self interaction. The
interaction may be multiplicative but it can take on other forms as well, e.g. you may need to
take logs of variables.
Dealing with Nonlinearity in variables. We will see that many nonlinear specifications can be
converted to linear form by performing transformations on the variables in the model. For
example, if Y is related to X by the equation E Yi Xi ( ) = + 2 and the relationship between the
variables is therefore nonlinear, we can define a new variable Z = X2 . The new variable Z is
then linearly related to Y, and OLS regression can be used to estimate the coefficients of the
model. There are numerous other cases where, given appropriate transformations of the
variables, nonlinear relationships can be converted into models for which coefficients can be
estimated using OLS. We’ll cover a few of the most important and common ones here, but there
are many others. Detecting nonlinearity and nonadditivity. The key question is whether the slope
of the relationship between an IV and a DV can be expected to vary depending on the context. •
The first step in detecting nonlinearity or nonadditivity is theoretical rather than technical. Once
the nature of the expected relationship is understood well enough to make a rough graph of it, the
technical work should begin. Hence, ask such questions as, can the slope of the relationship
between Xi and E(Y) be expected to have the same sign for all value.
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help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
RPMS TEMPLATE FOR SCHOOL YEAR 2023-2024 FOR TEACHER 1 TO TEACHER 3
6.2 Reflexivity, symmetry and transitivity (handout)
1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Reflexivity, symmetry and transitivity
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of December 4, 2017
2. Definition. Let R be a binary relation over X.
1. R is reflexive on X if ∀x∈X xRx.
2. R is symmetric on X if ∀x,y∈X (xRy → yRx).
3. R is transitive on X if ∀x,y,z∈X (xRy ∧ yRz → xRz).
Example. Let X be the set of all straight lines on a plane.
1. Relation isParallelTo is:
reflexive on X,
symmetric on X,
transitive on X.
2. Relation isPerpendicularTo is:
not reflexive on X,
symmetric on X,
not transitive on X.
3. Informal Example
Let X be a set of people.
1. Relation isASiblingOf on X is:
not reflexive on X,
symmetric on X,
not transitive on X.
It is not transitive, because
x isASiblingOf y ∧ y isASiblingOf x → x isASiblingOf x does not hold.
2. Relation isAnAncestorOf on X is:
not reflexive,
not symmetric,
transitive.
4. Exercise
Consider binary relations =Z, =Z, <Z, >Z, Z, Z, the total relation 1Z2 , the empty
relation ∅, and the relation | of divisibility.
1. For each of these relations, check if it is reflexive on Z, if it is symmetric on Z,
and if it is transitive on Z.
2. Graph these relations on the Cartesian plane. Observe that it is reflexivity and
symmetry of the relation is easy to recognize in a graph.
Exercise
Recall congruence modulo k: x ≡k y iff k|(y − x), for any fixed integer k > 1.
Make a discrete Cartesian graph of ≡3 which shows only those ordered pairs x, y
where x ≡3 y and x, y ∈ {−2..6}.
5. Proposition
≡k, is reflexive on Z, symmetric on Z and transitive on Z.
Proof. We will prove transitivity.
Take any x, y, z such that x ≡k y and y ≡k z.
Goal: x ≡k z.
By the definition of congruence mod k we have:
y − x = m · k, for some m ∈ Z and
z − y = n · k, for some n ∈ Z.
By adding these equations we obtain:
z − x = (m + n) · k, for some m, n ∈ Z.
So, z − x = l · k, for some l ∈ Z.
So, x ≡k z, by the definition of congruence mod k.
Exercise. Prove that ≡k is reflexive on Z and symmetric on Z.
6. Proposition
Let X is a set with n elements. Then, there are 2n(n−1) relations over X that are
reflexive on X.
Proof
Instead of counting relations reflexive on X it is enough to count subsets of the
Cartesian product X × X without the diagonal.
(Such relations and such subsets are in 1-1 correspondence – there is a bijection.)
The number of such relations is the same as the number of subsets of
(X × X) − { a, a : a ∈ X}.
This set contains n2 − n = n(n − 1) elements.
So the number of its subsets, and the number of relations in question is 2n(n−1).
7. Exercise. Assume that X is a set with n elements.
1. How many binary relations over X are there?
2. How many binary relations over X are reflexive on X?
3. How many binary relations over X are not reflexive on X?
4. How many binary relations over X are symmetric on X?
5. How many binary relations over X are not symmetric on X?
6. How many binary relations over X are reflexive on X and symmetric on X?
7. How many binary relations over X are reflexive on X and not symmetric on X?
8. How many binary relations over X are not reflexive on X and symmetric X?
9. How many binary relations over X are not reflexive on X and not symmetric on
X?
Hint: in order to answer the next question you may need to use answers to some of the
previous questions.
8. Claim. If R is symmetric on X and transitive on X then it is reflexive on X.
We will attempt to prove it.
Assume R is symmetric on X and transitive on X.
Take any x, y such that xRy.
By symmetry, yRx.
By transitivity, xRy ∧ yRx → xRx.
So, xRx.
We conclude that R is reflexive on X.
Is this conclusion correct? No. We proved only this:
If R is symmetric on X and transitive on X then it is reflexive on domain(R).
Exercise. Find R and X that disprove the Claim above.
Proposition
Let X = field(R). If R is symmetric on X and transitive on X then it is reflexive on X.
Exercise. Prove this proposition.