Proof Solutions: Inclass worksheet
... rational numbers is rational. 5) Typo Corrected: Theorem: Given an integer n, If n is even that n2 is divisible by 4. Proof: If n is even then n = 2a for some integer a (by definition of even) Consider n2= n*n = (2a)(2a) = 4a2 = 4d where d = a2 and d is an integer. Therefore n2 is divisible by 4 by ...
... rational numbers is rational. 5) Typo Corrected: Theorem: Given an integer n, If n is even that n2 is divisible by 4. Proof: If n is even then n = 2a for some integer a (by definition of even) Consider n2= n*n = (2a)(2a) = 4a2 = 4d where d = a2 and d is an integer. Therefore n2 is divisible by 4 by ...
Lecture slides (full content)
... then there exists a largest even number, m then let m’ = 2m then m’ is an even number, and larger than m # contradiction, since m is the largest even number then there are infinitely many even numbers # assuming ¬Q leads to contradiction, so Q. ...
... then there exists a largest even number, m then let m’ = 2m then m’ is an even number, and larger than m # contradiction, since m is the largest even number then there are infinitely many even numbers # assuming ¬Q leads to contradiction, so Q. ...
Pre-Greek math
... 3) Proofs by exhaustion by Archimedes is unsatisfactory as it is not causal (Rivaltus, 1615) ...
... 3) Proofs by exhaustion by Archimedes is unsatisfactory as it is not causal (Rivaltus, 1615) ...
Direct Proof and Counterexample II - H-SC
... Let Q denote the set of rational numbers. An irrational number is a number that is not rational. We will assume that there exist irrational numbers. ...
... Let Q denote the set of rational numbers. An irrational number is a number that is not rational. We will assume that there exist irrational numbers. ...
Quiz Game Midterm
... How do you check if an argument is valid using truth tables? Check all the rows in which the premises are all true. If the conclusion is always also true, it’s valid. If there’s a row where the premises are all true but the conclusion is false, it’s invalid. ...
... How do you check if an argument is valid using truth tables? Check all the rows in which the premises are all true. If the conclusion is always also true, it’s valid. If there’s a row where the premises are all true but the conclusion is false, it’s invalid. ...
(1) Find all prime numbers smaller than 100. (2) Give a proof by
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
A Short Introduction to the Idea of Proof.
... Office hour: 8-9pm, Sunday, in Sloan 155. Recitation: 10-11, Th, in Downs 11 Exercise 1.1. Take an 8x8 chessboard and remove one square from the upperright-hand corner. Can you cover it with 2x1 dominoes so that no two dominoes overlap? 2. What is a proof? Loosely defined, a proof is a rigorous math ...
... Office hour: 8-9pm, Sunday, in Sloan 155. Recitation: 10-11, Th, in Downs 11 Exercise 1.1. Take an 8x8 chessboard and remove one square from the upperright-hand corner. Can you cover it with 2x1 dominoes so that no two dominoes overlap? 2. What is a proof? Loosely defined, a proof is a rigorous math ...
Homework 1 (Due Tuesday April 5)
... Remember that a number r is said to be rational if r = ab where a and b are integers and b is nonzero. Recall that the integers are the counting numbers along with 0 and their negatives, i.e. {. . . , −2, −1, 0, 1, 2, . . .}. Exercise 1: If a and b are irrational numbers is a + b irrational? What ab ...
... Remember that a number r is said to be rational if r = ab where a and b are integers and b is nonzero. Recall that the integers are the counting numbers along with 0 and their negatives, i.e. {. . . , −2, −1, 0, 1, 2, . . .}. Exercise 1: If a and b are irrational numbers is a + b irrational? What ab ...
Propositional logic
... Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Defini ...
... Definition: an assignment to a set V of variables is a function s: V Æ {T,F}. Each assignment is inductively extended to apply to wffs. For wffs a and b • s(ÿa) = ÿs(a), • s(aŸb) = s(a) Ÿ s(b), • s(a⁄b) = s(a) ⁄ s(b), • s(afib) = s(a) fi s(b), • s(aÛb) = s(a) Û s(b), and • s(T) = T, s(F) = F, Defini ...
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.