ASSIGNMENT 3
... 1. Undefined Terms: Undefined terms are those terms that are accepted without any further definition. For example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined. 2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is acce ...
... 1. Undefined Terms: Undefined terms are those terms that are accepted without any further definition. For example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined. 2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is acce ...
Assignment # 3 : Solutions
... From definition of odd number, a=2n+1 and b=2k+1 for integers n, k. a+b = 2n+1+2k+1 = 2n + 2k+ 2 = 2*(n+k+1) Let integer s = n+k+1. Then we have a+b = 2*s. Therefore, from definition of even, a+b is even. 24.The problem with the given proof is that it “begs the question.” Explanation: Proof assume ...
... From definition of odd number, a=2n+1 and b=2k+1 for integers n, k. a+b = 2n+1+2k+1 = 2n + 2k+ 2 = 2*(n+k+1) Let integer s = n+k+1. Then we have a+b = 2*s. Therefore, from definition of even, a+b is even. 24.The problem with the given proof is that it “begs the question.” Explanation: Proof assume ...
Due Friday, 4/18/14 by 3 PM
... = . Since is a subset of , we can ask whether or not ∈ . Use the definition of the set (and remember that = ) to get a contradiction in either case, i.e., show that assuming ∈ leads to a contradiction, and assuming ∉ also leads to a contradiction. Since on ...
... = . Since is a subset of , we can ask whether or not ∈ . Use the definition of the set (and remember that = ) to get a contradiction in either case, i.e., show that assuming ∈ leads to a contradiction, and assuming ∉ also leads to a contradiction. Since on ...
![Primes in the Interval [2n, 3n]](http://s1.studyres.com/store/data/017368795_1-688e9118f5c89a0cb6fb7aedda33fc32-300x300.png)
Math 194, problem set #3
... Math 194, problem set #3 For discussion Tuesday, October 19, 2010 (1) For which integers n is (n3 − 3n2 + 4)/(2n − 1) an integer? (Andreescu & Gelca) (2) Is it possible to place 1995 different positive integers around a circle so that for any two adjacent numbers, the ratio of the larger to the smal ...
... Math 194, problem set #3 For discussion Tuesday, October 19, 2010 (1) For which integers n is (n3 − 3n2 + 4)/(2n − 1) an integer? (Andreescu & Gelca) (2) Is it possible to place 1995 different positive integers around a circle so that for any two adjacent numbers, the ratio of the larger to the smal ...
Proof Technique
... odd integers is n2. Proof: Let P(n) be the proposition that the sum of the first n odd integers is n2. Then to proof that P(n) is true for all n ≥ 1, we have to show that P(1) is true and P(k+1) is true if P(k) is true for k ≥ 1. Basic step: P(1) is true, because the sum of the first 1 odd integer ...
... odd integers is n2. Proof: Let P(n) be the proposition that the sum of the first n odd integers is n2. Then to proof that P(n) is true for all n ≥ 1, we have to show that P(1) is true and P(k+1) is true if P(k) is true for k ≥ 1. Basic step: P(1) is true, because the sum of the first 1 odd integer ...
Math 3000 Section 003 Intro to Abstract Math Homework 4
... fourth power is even. (c) The proof must conclude that the third multiple of that integer increased by 1 is an odd number. (d) i.ii. We know (already have proven) that an integer and its square have same parity. iii.v. These steps use the defintion of even and odd numbers. iv. This step uses substit ...
... fourth power is even. (c) The proof must conclude that the third multiple of that integer increased by 1 is an odd number. (d) i.ii. We know (already have proven) that an integer and its square have same parity. iii.v. These steps use the defintion of even and odd numbers. iv. This step uses substit ...
Chapter 1 Logic and Set Theory
... A proof in mathematics demonstrates the truth of certain statement. It is therefore natural to begin with a brief discussion of statements. A statement, or proposition, is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the ex ...
... A proof in mathematics demonstrates the truth of certain statement. It is therefore natural to begin with a brief discussion of statements. A statement, or proposition, is the content of an assertion. It is either true or false, but cannot be both true and false at the same time. For example, the ex ...
CS 103X: Discrete Structures Homework Assignment 2 — Solutions
... number p/q, we would have √6 = (p − 5q)/2q, which is a rational number. But we’ve proved the first part that 6 is irrational, which proves the result √ by contradiction. √ in √ For 2 + 6, the same method works, only we use the fact that 3 is irrational. Exercise 8 (20 points). Consider n lines in th ...
... number p/q, we would have √6 = (p − 5q)/2q, which is a rational number. But we’ve proved the first part that 6 is irrational, which proves the result √ by contradiction. √ in √ For 2 + 6, the same method works, only we use the fact that 3 is irrational. Exercise 8 (20 points). Consider n lines in th ...
Mathematical Induction Proof by Weak Induction
... then is an induction principle applicable? It is because we can restate the statement of the theorem as “for any natural number n and for any tree T , if n ≥ 0 and T is of order n then T has size n − 1.” Since no tree of order 0 exists, the implication in the paraphrase is true vacuously when n = 0. ...
... then is an induction principle applicable? It is because we can restate the statement of the theorem as “for any natural number n and for any tree T , if n ≥ 0 and T is of order n then T has size n − 1.” Since no tree of order 0 exists, the implication in the paraphrase is true vacuously when n = 0. ...
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.