MATH 363 Discrete Mathematics SOLUTIONS : Assignment 3 1
... p = 2m, then 8m3 = 2q 3 and thus, q 3 is even. Similarly, we have that q is even. But this √ is a contradiction to the assumption that p and q have no common factors. Therefore we conclude that 3 2 is irrational. 3. (4pt) Prove that either 2 · 3100 + 5 or 2 · 3100 + 6 is not a perfect square. Let n ...
... p = 2m, then 8m3 = 2q 3 and thus, q 3 is even. Similarly, we have that q is even. But this √ is a contradiction to the assumption that p and q have no common factors. Therefore we conclude that 3 2 is irrational. 3. (4pt) Prove that either 2 · 3100 + 5 or 2 · 3100 + 6 is not a perfect square. Let n ...
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... is tautology iff it is a theorem. The if part of the statement is the soundness theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. Propositional logic is sound with respect to truth-value semantics. Proof. ...
... is tautology iff it is a theorem. The if part of the statement is the soundness theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. Propositional logic is sound with respect to truth-value semantics. Proof. ...
gödel`s completeness theorem with natural language formulas
... Proof . This follows immediately from the construction and properties (1) and (2). qed(3) (4) For every formula B, “not B” ∈ H iff B 6∈ H . Proof . Consider B = Fm . Assume that “not B” ∈ H . Assume for a contradiction that also B ∈ H . Choose a natural number n such that B, “not B”∈ {A1 , . . . , A ...
... Proof . This follows immediately from the construction and properties (1) and (2). qed(3) (4) For every formula B, “not B” ∈ H iff B 6∈ H . Proof . Consider B = Fm . Assume that “not B” ∈ H . Assume for a contradiction that also B ∈ H . Choose a natural number n such that B, “not B”∈ {A1 , . . . , A ...
2015Khan-What is Math-anOverview-IJMCS-2015
... They are not invented by us but rather discovered. Formalists on the other hand believe that there are no such things as mathematical objects. Mathematics consists of definitions, axioms and theorems invented by mathematicians and have no meaning in themselves except that which we ascribe to them. T ...
... They are not invented by us but rather discovered. Formalists on the other hand believe that there are no such things as mathematical objects. Mathematics consists of definitions, axioms and theorems invented by mathematicians and have no meaning in themselves except that which we ascribe to them. T ...
Resources - CSE, IIT Bombay
... − As opposed to interrogative statements (questions) or imperative statements (request, order) ...
... − As opposed to interrogative statements (questions) or imperative statements (request, order) ...
Methods of Proof
... Theorem (by Euclid): There are infinitely many prime numbers. Proof. Assume there are a finite number of primes List them as follows: p1, p2 …, pn. Consider the number q = p1p2 … pn + 1 ...
... Theorem (by Euclid): There are infinitely many prime numbers. Proof. Assume there are a finite number of primes List them as follows: p1, p2 …, pn. Consider the number q = p1p2 … pn + 1 ...
Number theory and proof techniques
... n and asks to find factors of it Integers r and s such that rs=n How would you do it? Can try all integers from 1 to n Can try all integers from 1 to sqrt(n) Can try all primes from 1 to sqrt(n) ...
... n and asks to find factors of it Integers r and s such that rs=n How would you do it? Can try all integers from 1 to n Can try all integers from 1 to sqrt(n) Can try all primes from 1 to sqrt(n) ...
Full text
... Thus, given any number from the collection j 3, 9, 27, 13, 39, 39 \ there exists an/?GT such that/?M is the given number. The following theorem verifies that every positive integer can be obtained in this manner. Before stating the theorem, the following conventions are adopted. The set qf non-negat ...
... Thus, given any number from the collection j 3, 9, 27, 13, 39, 39 \ there exists an/?GT such that/?M is the given number. The following theorem verifies that every positive integer can be obtained in this manner. Before stating the theorem, the following conventions are adopted. The set qf non-negat ...
Prime Numbers - Winchester College
... the whole together with the square on the straight line between the points of section is equal to the square on the ...
... the whole together with the square on the straight line between the points of section is equal to the square on the ...
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.