n - Stanford University
... A Simple Proof by Contradiction Theorem: If n2 is even, then n is even. Proof: By contradiction; assume n2 is even but n is odd. Since n is odd, n = 2k + 1 for some integer k. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Now, let m = 2k2 + 2k. Then n2 = 2m + 1, so by definition n2 is odd. ...
... A Simple Proof by Contradiction Theorem: If n2 is even, then n is even. Proof: By contradiction; assume n2 is even but n is odd. Since n is odd, n = 2k + 1 for some integer k. Then n2 = (2k + 1)2 = 4k2 + 4k + 1 = 2(2k2 + 2k) + 1. Now, let m = 2k2 + 2k. Then n2 = 2m + 1, so by definition n2 is odd. ...
Solution - Stony Brook Mathematics
... than x (since x/2 < x by the multiplicative law for inequalities), a contradiction. Therefore, there does not exist a smallest positive real number. Problem 4. Use induction to prove Bernouilli’s inequality : (1 + x)n ≥ 1 + nx for all non-negative integers n and real numbers x > −1. Sol. Fix some re ...
... than x (since x/2 < x by the multiplicative law for inequalities), a contradiction. Therefore, there does not exist a smallest positive real number. Problem 4. Use induction to prove Bernouilli’s inequality : (1 + x)n ≥ 1 + nx for all non-negative integers n and real numbers x > −1. Sol. Fix some re ...
Test 2A (Sec. 2-1, 2-2, 2-5, 2-6) Multiple Choice Identify the choice
... Plan: Since both pairs of angle measures add to 90 , use substitution to show that the sums of both pairs are equal. Since m = m , use substitution again to show that sums of the other pairs are equal. Use the Subtraction Property of Equality to conclude that m = m . Complete the proof. ...
... Plan: Since both pairs of angle measures add to 90 , use substitution to show that the sums of both pairs are equal. Since m = m , use substitution again to show that sums of the other pairs are equal. Use the Subtraction Property of Equality to conclude that m = m . Complete the proof. ...
Math 344 Winter 07 Group Theory Part 2: Subgroups and Isomorphism
... isomorphic! And they give a way to check whether we need to go to the trouble of creating an isomorphism. When asked to prove/disprove whether two groups are isomorphic, check to see if they have the same properties? • Commutative? (if one is, they both need to be) • Cyclic (if one is, they both nee ...
... isomorphic! And they give a way to check whether we need to go to the trouble of creating an isomorphism. When asked to prove/disprove whether two groups are isomorphic, check to see if they have the same properties? • Commutative? (if one is, they both need to be) • Cyclic (if one is, they both nee ...
Surprising Connections between Partitions and Divisors
... “partitions” of a given number as sums of positive integers. For example, the seven partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1. The “partition function” p(n) is defined as the number of partitions of n. Thus, p(5) = 7. Prime numbers and divisors ...
... “partitions” of a given number as sums of positive integers. For example, the seven partitions of 5 are 5, 4 + 1, 3 + 2, 3 + 1 + 1, 2 + 2 + 1, 2 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1. The “partition function” p(n) is defined as the number of partitions of n. Thus, p(5) = 7. Prime numbers and divisors ...
The Nature of Mathematics
... Theorem 4.1 (Distributive law). For sets A, B and C, prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Proof. Let x ∈ A ∩ (B ∪ C). Then x ∈ A and x ∈ B ∪ C. Case I. Let x ∈ B. Then x ∈ A ∩ B =⇒ x ∈ (A ∩ B) ∪ (A ∩ C). Case II. Let x ∈ C. Then x ∈ A ∩ C =⇒ x ∈ (A ∩ B) ∪ (A ∩ C). Thus A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ ...
... Theorem 4.1 (Distributive law). For sets A, B and C, prove that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). Proof. Let x ∈ A ∩ (B ∪ C). Then x ∈ A and x ∈ B ∪ C. Case I. Let x ∈ B. Then x ∈ A ∩ B =⇒ x ∈ (A ∩ B) ∪ (A ∩ C). Case II. Let x ∈ C. Then x ∈ A ∩ C =⇒ x ∈ (A ∩ B) ∪ (A ∩ C). Thus A ∩ (B ∪ C) ⊆ (A ∩ B) ∪ ...
ppt - UBC Computer Science
... ∀x ∈ D, P(x) → Q(x) This is a special case of the previous formula The textbook calls this (and only this) a direct proof. The proof looks like this: Proof: o Consider an unspecified element k of D. o Assume that P(k) is true. o Use this and properties of the element of D to verify that the ...
... ∀x ∈ D, P(x) → Q(x) This is a special case of the previous formula The textbook calls this (and only this) a direct proof. The proof looks like this: Proof: o Consider an unspecified element k of D. o Assume that P(k) is true. o Use this and properties of the element of D to verify that the ...
predicate
... holds in directed graphs iff there is a path from u to v. • Proof. By contradiction. Suppose there is such a formula. Let n be the formula expressing that there is a path from c to c’ ...
... holds in directed graphs iff there is a path from u to v. • Proof. By contradiction. Suppose there is such a formula. Let n be the formula expressing that there is a path from c to c’ ...
Slides for Rosen, 5th edition
... • Mathematics is much more than that: Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts. ...
... • Mathematics is much more than that: Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts. ...
PROBLEM SET 01 - Proofs and induction 1 Proofs 2 Induction
... Exercise 2.6. Consider a game in which two players take turns removing any positive number of matches they want from one of two piles of matches. The player who removes the last match wins the game. Show that if the two piles contain the same number of matches initially, the second player can always ...
... Exercise 2.6. Consider a game in which two players take turns removing any positive number of matches they want from one of two piles of matches. The player who removes the last match wins the game. Show that if the two piles contain the same number of matches initially, the second player can always ...
PDF
... as the theorem states. For example, the sum of the first four cubes is 1 + 9 + 27 + 64 = 100. This is also equal to 1 + 3 + 5 + 7 + 9 + 11 + 13 + 17 + 19 = 100. The square root of 100 is 10, the fourth triangular number, and indeed 10 = 1 + 2 + 3 + ...
... as the theorem states. For example, the sum of the first four cubes is 1 + 9 + 27 + 64 = 100. This is also equal to 1 + 3 + 5 + 7 + 9 + 11 + 13 + 17 + 19 = 100. The square root of 100 is 10, the fourth triangular number, and indeed 10 = 1 + 2 + 3 + ...
Theorem If p is a prime number which has remainder 1 when
... Proof the class came up with after much discussion: First we need to define even and odd. Definition: We define an integer n to be even if there exists an integer k such that n = 2k. We define an integer n to be odd if there’s an integer k such that n = 2k + 1. Proof of theorem: Write n2 + n = n(n ...
... Proof the class came up with after much discussion: First we need to define even and odd. Definition: We define an integer n to be even if there exists an integer k such that n = 2k. We define an integer n to be odd if there’s an integer k such that n = 2k + 1. Proof of theorem: Write n2 + n = n(n ...
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.