lecture notes on mathematical induction
... instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical induction. Alternately, the principle of mathematical induction is a key ingredient in any axiomatic characterization of the natural numbe ...
... instance in terms of sets, as alluded to earlier in the course) needs to incorporate, either implicitly or (more often) explicitly, the principle of mathematical induction. Alternately, the principle of mathematical induction is a key ingredient in any axiomatic characterization of the natural numbe ...
Math 318 Class notes
... g f = {( a, c) ∈ A × C : ∃b ∈ B, ( a, b) ∈ f , (b, c) ∈ g} Definition. Given a function f : A → B and function g : B → A. g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. f is injective iff it has a left inverse. Proof. ⇒ Write B = ra ...
... g f = {( a, c) ∈ A × C : ∃b ∈ B, ( a, b) ∈ f , (b, c) ∈ g} Definition. Given a function f : A → B and function g : B → A. g is called a left inverse of f if g f = id A ; g is called a right inverse of f if f g = id B . Proposition 3.11. f is injective iff it has a left inverse. Proof. ⇒ Write B = ra ...
Mathematical induction Elad Aigner-Horev
... form. Below we shall consider examples that will clarify this point. It is clear that anything that can be proved using weak induction can be proved using strong induction. The converse holds as well. In fact, these two techniques are equivalent. Theorem 2.3. The principles of weak and strong induct ...
... form. Below we shall consider examples that will clarify this point. It is clear that anything that can be proved using weak induction can be proved using strong induction. The converse holds as well. In fact, these two techniques are equivalent. Theorem 2.3. The principles of weak and strong induct ...
Binary aggregation with integrity constraints Grandi, U. - UvA-DARE
... knowledge SWFs defined on pair judgments have not yet been studied in the literature, and we now prove a possibility result concerning this class of procedures. Proposition 5.2.2. There exists a SWF defined on pair judgments that satisfies the Pareto condition, May’s neutrality, independence and pos ...
... knowledge SWFs defined on pair judgments have not yet been studied in the literature, and we now prove a possibility result concerning this class of procedures. Proposition 5.2.2. There exists a SWF defined on pair judgments that satisfies the Pareto condition, May’s neutrality, independence and pos ...
Proof Theory: From Arithmetic to Set Theory
... A first order language L is specified by its non-logical symbols. These symbols are separated into three groups: LC , LF , and LR . LC is the set of constant symbols, LF is the set of function symbols, and LR is the set of relation symbols. Each function symbol f ∈ LF also comes equipped with an ari ...
... A first order language L is specified by its non-logical symbols. These symbols are separated into three groups: LC , LF , and LR . LC is the set of constant symbols, LF is the set of function symbols, and LR is the set of relation symbols. Each function symbol f ∈ LF also comes equipped with an ari ...
CHAPTER 1 The main subject of Mathematical Logic is
... In a given context we shall adopt the following convention. Once a formula has been introduced as A(x), i.e., A with a designated variable x, we write A(r) for A[x := r], and similarly with more variables. 1.1.3. Subformulas. Unless stated otherwise, the notion of subformula will be that defined by ...
... In a given context we shall adopt the following convention. Once a formula has been introduced as A(x), i.e., A with a designated variable x, we write A(r) for A[x := r], and similarly with more variables. 1.1.3. Subformulas. Unless stated otherwise, the notion of subformula will be that defined by ...
MMConceptualComputationalRemainder
... the maximum of the set of common divisors of the two numbers, and a set of numbers has only one maximum. I have shown my students this proof many times, but they almost never reproduce it on an examination. ...
... the maximum of the set of common divisors of the two numbers, and a set of numbers has only one maximum. I have shown my students this proof many times, but they almost never reproduce it on an examination. ...
Strong Completeness for Iteration
... detail later). Such maps can be viewed as arrows in the Kleisli category of the monad T which yields semantics of sequential composition as Kleisli composition. Alternatively, a map X → T X can be viewed as a T -coalgebra which leads to a (coalgebraic) modal logic of T -computations. Other construct ...
... detail later). Such maps can be viewed as arrows in the Kleisli category of the monad T which yields semantics of sequential composition as Kleisli composition. Alternatively, a map X → T X can be viewed as a T -coalgebra which leads to a (coalgebraic) modal logic of T -computations. Other construct ...
Induction
... We need to justify the Principle of Mathematical Induction by proving Theorem 3.1. To do this we need to make a new assumption about the natural numbers. It turns out that the Principle of Mathematical Induction is equivalent to this new assumption. Theorem 3.14. Let P (n) be a statement for each n ...
... We need to justify the Principle of Mathematical Induction by proving Theorem 3.1. To do this we need to make a new assumption about the natural numbers. It turns out that the Principle of Mathematical Induction is equivalent to this new assumption. Theorem 3.14. Let P (n) be a statement for each n ...
Logics of Truth - Project Euclid
... to Scott [10] and Aczel [1]. The central notion is AczePs concept of a Frege structure. These structures are models of the Lambda Calculus together with two distinguished subsets —a set of propositions and a subset of this set called truths. In addition, such structures come equipped with the usual ...
... to Scott [10] and Aczel [1]. The central notion is AczePs concept of a Frege structure. These structures are models of the Lambda Calculus together with two distinguished subsets —a set of propositions and a subset of this set called truths. In addition, such structures come equipped with the usual ...
Ordered Groups: A Case Study In Reverse Mathematics 1 Introduction
... Weak König’s Lemma. Every infinite binary branching tree has a path. The second subsystem of Z2 is called W KL0 and contains the axioms of RCA0 plus Weak König’s Lemma. Because the effective version of Weak König’s Lemma fails, W KL0 is strictly stronger than RCA0 . The best intuition for W KL0 ...
... Weak König’s Lemma. Every infinite binary branching tree has a path. The second subsystem of Z2 is called W KL0 and contains the axioms of RCA0 plus Weak König’s Lemma. Because the effective version of Weak König’s Lemma fails, W KL0 is strictly stronger than RCA0 . The best intuition for W KL0 ...
classden
... continuous functions from D to D. This guarantees that any object d ∈ D is also a function d : D → D and hence that it is meaningful to talk about d(d). Scott domains thus support the interpretation of self-application and in fact are essential for the interpretation of functional languages which ar ...
... continuous functions from D to D. This guarantees that any object d ∈ D is also a function d : D → D and hence that it is meaningful to talk about d(d). Scott domains thus support the interpretation of self-application and in fact are essential for the interpretation of functional languages which ar ...
Axiomatic Set Teory P.D.Welch.
... the language L with other predicate symbols A⃗ = A , A , . . .; if this is done we denote the appropriate language by L A⃗. We use the binary predicate symbol P as a relation to be interpreted as membership: “v P v ” will be interpreted as “v is a member of v ” etc. We often use other letters ...
... the language L with other predicate symbols A⃗ = A , A , . . .; if this is done we denote the appropriate language by L A⃗. We use the binary predicate symbol P as a relation to be interpreted as membership: “v P v ” will be interpreted as “v is a member of v ” etc. We often use other letters ...
A Paedagogic Example of Cut-Elimination
... upper bound of {y}. By transitivity, the soundness of ∩:left and ∪:right follows. If x ≤ y and x ≤ z (i.e., x is a lower bound of {y, z}), then by the definition of greatest lower bound, x is also less than y ∩ z which means that ∩:right is sound. Similarly for ∪:left. The rules comm:left and comm:r ...
... upper bound of {y}. By transitivity, the soundness of ∩:left and ∪:right follows. If x ≤ y and x ≤ z (i.e., x is a lower bound of {y, z}), then by the definition of greatest lower bound, x is also less than y ∩ z which means that ∩:right is sound. Similarly for ∪:left. The rules comm:left and comm:r ...
Ans - Logic Matters
... containing the parameter α and ξ is a variable (new to ϕ), then ∀ξϕ0 is a wff (where ϕ0 results from ϕ by substituting ξ for some or all occurrences of α). (f) Again you need to consider cases. 1. Given an expression e, test to see if it is an atomic wff, which we can do effectively by part (d). If ...
... containing the parameter α and ξ is a variable (new to ϕ), then ∀ξϕ0 is a wff (where ϕ0 results from ϕ by substituting ξ for some or all occurrences of α). (f) Again you need to consider cases. 1. Given an expression e, test to see if it is an atomic wff, which we can do effectively by part (d). If ...
Chapter 2 - Princeton University Press
... greatly enjoy its computations. It is those few who may choose to study beyond the first course (i.e., beyond this book) and become logicians. I would expect that most students taking a first course in mathematical logic are simply liberal arts students fulfilling a mathematics requirement and seeki ...
... greatly enjoy its computations. It is those few who may choose to study beyond the first course (i.e., beyond this book) and become logicians. I would expect that most students taking a first course in mathematical logic are simply liberal arts students fulfilling a mathematics requirement and seeki ...
How to Prove Properties by Induction on Formulas
... Worcester Polytechnic Institute January 23, 2010 Formulas are finite, recursively generated objects, and therefore we can prove properties about all formulas using structural induction. “Structural induction” refers to induction principles that work directly on the structure of recursively generated ...
... Worcester Polytechnic Institute January 23, 2010 Formulas are finite, recursively generated objects, and therefore we can prove properties about all formulas using structural induction. “Structural induction” refers to induction principles that work directly on the structure of recursively generated ...
Introductory Mathematics
... is different from all the other previous natural numbers, so that the process never ends. Conversely, every natural number is a successor of the previous number, except 0. Notice that these numbers have an order starting from 0, followed by 1, followed by 2 and so on. We denote this by using the sym ...
... is different from all the other previous natural numbers, so that the process never ends. Conversely, every natural number is a successor of the previous number, except 0. Notice that these numbers have an order starting from 0, followed by 1, followed by 2 and so on. We denote this by using the sym ...
Hilbert Type Deductive System for Sentential Logic, Completeness
... Sketch of Proof: Since, by Theorem 5, α→ β |– ¬β→¬ α and ¬α→ β |– ¬β→¬¬α , it is sufficient to show that ¬β→¬ α, ¬β→¬¬α |– β. Now, ¬β→¬ α, ¬β→¬¬α, ¬β is a contradictory premise set, since both ¬α and ¬¬α are provable from it. Hence, ¬β→¬ α, ¬β→¬¬α |– ¬¬ β. Since ¬¬β |– β, we get the desired conclus ...
... Sketch of Proof: Since, by Theorem 5, α→ β |– ¬β→¬ α and ¬α→ β |– ¬β→¬¬α , it is sufficient to show that ¬β→¬ α, ¬β→¬¬α |– β. Now, ¬β→¬ α, ¬β→¬¬α, ¬β is a contradictory premise set, since both ¬α and ¬¬α are provable from it. Hence, ¬β→¬ α, ¬β→¬¬α |– ¬¬ β. Since ¬¬β |– β, we get the desired conclus ...
A short article for the Encyclopedia of Artificial Intelligence: Second
... implementation of theorem provers. In higher-order logics with λ-abstractions within terms (such as the Simple Theory of Types), bound variables are handled by the logic via α, β, and η-conversion (in comparison to, say, a first-order system such as Prolog where bound variables are handled only by p ...
... implementation of theorem provers. In higher-order logics with λ-abstractions within terms (such as the Simple Theory of Types), bound variables are handled by the logic via α, β, and η-conversion (in comparison to, say, a first-order system such as Prolog where bound variables are handled only by p ...
Sample pages 1 PDF
... is called a semigroup, and if ◦ is additionally invertible, then A is said to be a group. It is provable that a group (G, ◦) in this sense contains exactly one unit element, that is, an element e such that x ◦ e = e ◦ x = x for all x ∈ G, also called a neutral element. A well-known example is the gr ...
... is called a semigroup, and if ◦ is additionally invertible, then A is said to be a group. It is provable that a group (G, ◦) in this sense contains exactly one unit element, that is, an element e such that x ◦ e = e ◦ x = x for all x ∈ G, also called a neutral element. A well-known example is the gr ...
Induction
... But this directly contradicts the fact that P(m − 1) =⇒ P(m). It may seem as though we just proved the induction axiom. But what we have actually done is to show that the induction axiom follows from another axiom, which was used implicitly in defining “the first m for which P(m) is false.” We note ...
... But this directly contradicts the fact that P(m − 1) =⇒ P(m). It may seem as though we just proved the induction axiom. But what we have actually done is to show that the induction axiom follows from another axiom, which was used implicitly in defining “the first m for which P(m) is false.” We note ...
Partition of a Set which Contains an Infinite Arithmetic (Respectively
... (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression. Introduction. First, in this article we build sets which have the following property: for any par ...
... (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression. Introduction. First, in this article we build sets which have the following property: for any par ...
Introduction to Logic for Computer Science
... trying to symbolise the whole of mathematics could be disastrous as then it would become quite impossible to even read and understand mathematics, since what is presented usually as a one page proof could run into several pages. But at least in principle it can be done. Since the latter half of the ...
... trying to symbolise the whole of mathematics could be disastrous as then it would become quite impossible to even read and understand mathematics, since what is presented usually as a one page proof could run into several pages. But at least in principle it can be done. Since the latter half of the ...
Distance, Ruler Postulate and Plane Separation Postulate
... 1. The metric function that measures the distance 2. A coordinate function that tells us how to place the ruler ● When we place a ruler to measure a distance we can use any section of the ruler to measure ● The easiest way to place the ruler is to put it down so that 0 is at the left end of the segm ...
... 1. The metric function that measures the distance 2. A coordinate function that tells us how to place the ruler ● When we place a ruler to measure a distance we can use any section of the ruler to measure ● The easiest way to place the ruler is to put it down so that 0 is at the left end of the segm ...
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.