100Salmon
... i) Either nature is uniform or it is not. ii) If nature is uniform, then we should use scientific method. iii) If nature is not uniform, then we are “out of luck” anyway. SO: iv) We have much to gain, nothing to lose, in assuming nature uniform. [note similarity to James, pp. 106-14] ...
... i) Either nature is uniform or it is not. ii) If nature is uniform, then we should use scientific method. iii) If nature is not uniform, then we are “out of luck” anyway. SO: iv) We have much to gain, nothing to lose, in assuming nature uniform. [note similarity to James, pp. 106-14] ...
Notes: Chapter 2 Section 2.3: The Pigeonhole Principle Definition 1
... Definition 1. If more than n · k objects are placed into n categories, then there exists a category with more than k objects in it. Example 1. A drawer with 8 black socks and 11 blue socks. It’s dark and you can’t tell them apart. How many must you take to guarantee 2 of the same color? 2 black? 2 b ...
... Definition 1. If more than n · k objects are placed into n categories, then there exists a category with more than k objects in it. Example 1. A drawer with 8 black socks and 11 blue socks. It’s dark and you can’t tell them apart. How many must you take to guarantee 2 of the same color? 2 black? 2 b ...
My Favourite Proofs of the Infinitude of Primes Chris Almost
... The following proofs depend on the fact that the positive integers grow without bound, and on the following simple lemma. Lemma 1. If n ∈ Z and |n| = 6 1 then there is prime number that divides n. Proof. Every prime divides 0, so suppose n is minimal such that |n| = 6 1 and there is no prime dividin ...
... The following proofs depend on the fact that the positive integers grow without bound, and on the following simple lemma. Lemma 1. If n ∈ Z and |n| = 6 1 then there is prime number that divides n. Proof. Every prime divides 0, so suppose n is minimal such that |n| = 6 1 and there is no prime dividin ...
Every prime of the form 4k+1 is the sum of two perfect squares
... Numbers by G. H. Hardy and E. M. Wright. My copy is battered 4th edition, published by Oxford University Press. The relevant theorems are 82, 86 and the third proof of Theorem 366 in section 20.4 We will assume familiarity with the integers modulo p where p is a prime number, as it will be throughou ...
... Numbers by G. H. Hardy and E. M. Wright. My copy is battered 4th edition, published by Oxford University Press. The relevant theorems are 82, 86 and the third proof of Theorem 366 in section 20.4 We will assume familiarity with the integers modulo p where p is a prime number, as it will be throughou ...
Math 232 - Discrete Math Notes 2.1 Direct Proofs and
... Thus a2 is divisible by 5. That this implies a is divisible by 5. Therefore, a = 5k for some integer k. Then 5b2 = (5k)2 = 25k 2 → b2 = 5k 2 . That implies that b2 and therefore √ b is divisible by 5. But then a, b share the factor 5. This is a contradiction. Therefore, 5 is ...
... Thus a2 is divisible by 5. That this implies a is divisible by 5. Therefore, a = 5k for some integer k. Then 5b2 = (5k)2 = 25k 2 → b2 = 5k 2 . That implies that b2 and therefore √ b is divisible by 5. But then a, b share the factor 5. This is a contradiction. Therefore, 5 is ...
Document
... • Direct proof: Assume p is true, and prove q. • Indirect proof: Assume q, and prove p. • Vacuous proof: Prove p by itself. • Trivial proof: Prove q by itself. • Proof by cases: – to prove pq, where p and (p1 p2) are equiv. - show (p1q) and (p2q). ...
... • Direct proof: Assume p is true, and prove q. • Indirect proof: Assume q, and prove p. • Vacuous proof: Prove p by itself. • Trivial proof: Prove q by itself. • Proof by cases: – to prove pq, where p and (p1 p2) are equiv. - show (p1q) and (p2q). ...
HOMEWORK 2 1. P63, Ex. 1 Proof. We prove it by contradiction
... That is to say, p2 is a multiple of 3. We may classify p in the following cases, p = 3k, p = 3k + 1, p = 3k + 2, for integers k. In the latter two cases p2 = 9k 2 + 12k + 4, p2 = 9k 2 + 12k + 4, which are not multiples of 3. So p = 3k. Then we have q 2 = 3k 2 , so by the reasoning above, q is a mult ...
... That is to say, p2 is a multiple of 3. We may classify p in the following cases, p = 3k, p = 3k + 1, p = 3k + 2, for integers k. In the latter two cases p2 = 9k 2 + 12k + 4, p2 = 9k 2 + 12k + 4, which are not multiples of 3. So p = 3k. Then we have q 2 = 3k 2 , so by the reasoning above, q is a mult ...
On three consecutive primes
... Introduction The following result is due to Ishikawa [5]. Theorem 1. ...
... Introduction The following result is due to Ishikawa [5]. Theorem 1. ...
Direct Proofs (continued)
... In this lecture Number Theory ● Rational numbers ● Divisibility Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample ...
... In this lecture Number Theory ● Rational numbers ● Divisibility Proofs ● Direct proofs (cont.) ● Common mistakes in proofs ● Disproof by counterexample ...
rendering
... 1. Introduction We will exemplify here how to number, label and refer to sections, theorems etc. in a LATEX paper. Along the way, we will prove a remarkable result — see Section 3. 2. Preliminaries We want to prove that the even numbers form a ring with the operations of addition and multiplication. ...
... 1. Introduction We will exemplify here how to number, label and refer to sections, theorems etc. in a LATEX paper. Along the way, we will prove a remarkable result — see Section 3. 2. Preliminaries We want to prove that the even numbers form a ring with the operations of addition and multiplication. ...
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.