Lecture 3. Mathematical Induction
... Induction in natural sciences cannot be absolute, because it is based on a very large but finite number of observations and experiments. We know that in the process of the evolution of such sciences as physics or biology, all the fundamental laws from time to time have been revised. For example, New ...
... Induction in natural sciences cannot be absolute, because it is based on a very large but finite number of observations and experiments. We know that in the process of the evolution of such sciences as physics or biology, all the fundamental laws from time to time have been revised. For example, New ...
2.1-2.3: Reasoning in Geometry
... Definition of a Vehicle “Anything that has wheels and moves people from place to place.” Not all definitions may be precise, so when creating or following one, read carefully! ...
... Definition of a Vehicle “Anything that has wheels and moves people from place to place.” Not all definitions may be precise, so when creating or following one, read carefully! ...
Greatest Common Divisors and Linear Combinations Let a and b be
... Let a and b be positive integers. The greatest common divisor of a and b (“gcd(a, b)”) has a close and very useful connection to things called “linear combinations” of a and b (defined below). This handout will explain the connection. Definition 1 Let a and b be positive integers. A linear combinati ...
... Let a and b be positive integers. The greatest common divisor of a and b (“gcd(a, b)”) has a close and very useful connection to things called “linear combinations” of a and b (defined below). This handout will explain the connection. Definition 1 Let a and b be positive integers. A linear combinati ...
Proofs
... Restate implication as p is prime, and there does not exist a prime which is greater. 1. Form a product r = 2 · 3 · 5 · ... p) (e.g. r is the product of all primes) 2. If we divide r+1 by any prime, it will have remainder 1 3. r+1 is prime, since any number not divisible by any prime which is less m ...
... Restate implication as p is prime, and there does not exist a prime which is greater. 1. Form a product r = 2 · 3 · 5 · ... p) (e.g. r is the product of all primes) 2. If we divide r+1 by any prime, it will have remainder 1 3. r+1 is prime, since any number not divisible by any prime which is less m ...
Full text
... Proof. We have only to observe that the right factor of (3) cannot be positive unless equation (2) holds. Here we are using an idea of Putnam [ 3 ] . It will be seen that the polynomial (3) also gives certain negative values. This is unavoidable. It is easy to prove that a polynomial which takes onl ...
... Proof. We have only to observe that the right factor of (3) cannot be positive unless equation (2) holds. Here we are using an idea of Putnam [ 3 ] . It will be seen that the polynomial (3) also gives certain negative values. This is unavoidable. It is easy to prove that a polynomial which takes onl ...
6.042J Lecture 02: Solutions
... Now we justify the claim that if an is even, so is a. There is a simple proof by contradiction: suppose to the contrary that a is odd. It’s a familiar (and easily verified1 ) fact that the product of two odd numbers is odd, from which it follows that the product of any finite number of odd numbers is ...
... Now we justify the claim that if an is even, so is a. There is a simple proof by contradiction: suppose to the contrary that a is odd. It’s a familiar (and easily verified1 ) fact that the product of two odd numbers is odd, from which it follows that the product of any finite number of odd numbers is ...
Sets, Numbers, and Logic
... no common factors, i.e., that the representation is in lowest terms. You can’t always write every fraction in lowest terms, however: it makes calculation too hard. For instance, any two rational numbers can be added only because they are “commensurable”, that is, can be written so they have a common ...
... no common factors, i.e., that the representation is in lowest terms. You can’t always write every fraction in lowest terms, however: it makes calculation too hard. For instance, any two rational numbers can be added only because they are “commensurable”, that is, can be written so they have a common ...
PPT
... Informal Definition: “Independence of Irrelevant Alternatives” (IIA) Philosopher Sidney Morgenbesser is ordering dessert. The waiter says they have apple and blueberry pie. Morgenbesser asks for apple. The waiter comes back out and says “Oh, we have cherry as well!” “In that case,” says Morgenbesse ...
... Informal Definition: “Independence of Irrelevant Alternatives” (IIA) Philosopher Sidney Morgenbesser is ordering dessert. The waiter says they have apple and blueberry pie. Morgenbesser asks for apple. The waiter comes back out and says “Oh, we have cherry as well!” “In that case,” says Morgenbesse ...
CS 399: Constructive Logic Final Exam (Sample Solution) Name Instructions
... Unequality: We will say x and y are unequal if ¬(x =N y). In the next two problems you will show that x 6= y if and only if x and y are unequal. The proof terms for =N are given on page 57 of the Pfenning notes. (B) Prove that ∀x ∈ nat. ∀y ∈ nat.¬(x =N y) ⊃ x 6= y. Give your ...
... Unequality: We will say x and y are unequal if ¬(x =N y). In the next two problems you will show that x 6= y if and only if x and y are unequal. The proof terms for =N are given on page 57 of the Pfenning notes. (B) Prove that ∀x ∈ nat. ∀y ∈ nat.¬(x =N y) ⊃ x 6= y. Give your ...
MIDTERM REVIEW FOR MATH 500 1. The limit Define limn→∞ an
... the completeness Axiom establish some surprising properties of the real numbers, for instance, the Archimedean property of real numbers, and the “approximation” of any numbers in R by integers, and approximation of real numbers by rational numbers. Here we give an example to establish the monotone c ...
... the completeness Axiom establish some surprising properties of the real numbers, for instance, the Archimedean property of real numbers, and the “approximation” of any numbers in R by integers, and approximation of real numbers by rational numbers. Here we give an example to establish the monotone c ...
9.6 Mathematical Induction
... difficult to win once you get the hang of it, but it takes a while to move all the washers even when you know what you are doing. A mathematician, presented with this game, wants to figure out the minimum number of moves required to win the game— not because of impatience, but because it is an inter ...
... difficult to win once you get the hang of it, but it takes a while to move all the washers even when you know what you are doing. A mathematician, presented with this game, wants to figure out the minimum number of moves required to win the game— not because of impatience, but because it is an inter ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.