MATH 312H–FOUNDATIONS
... Principle of mathematical induction. Let for each natural number n be given a statement An . Then if • Base of induction: the statement A1 is true. • Induction step: If the statement An is true then An+1 holds as well. the conclusion of the mathematical induction is that all statements An are true. ...
... Principle of mathematical induction. Let for each natural number n be given a statement An . Then if • Base of induction: the statement A1 is true. • Induction step: If the statement An is true then An+1 holds as well. the conclusion of the mathematical induction is that all statements An are true. ...
Course Description
... BSCS-411 — Discrete Mathematics (Mathematical Induction) — Week 6 Mathematical Induction • A powerful, rigorous technique for proving that a predicate P(n) is true for every natural number n, no matter how large. • Essentially a “domino effect” principle. • Based on a predicate-logic inference rule: ...
... BSCS-411 — Discrete Mathematics (Mathematical Induction) — Week 6 Mathematical Induction • A powerful, rigorous technique for proving that a predicate P(n) is true for every natural number n, no matter how large. • Essentially a “domino effect” principle. • Based on a predicate-logic inference rule: ...
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction
... The provable formulas for which we have a formal proof are called consequences of the axioms, or theorem, or just simple provable formulas. When building a proof system we choose not only axioms of the system, but also specific rules of inference. The choice of rules is often linked, as was the choi ...
... The provable formulas for which we have a formal proof are called consequences of the axioms, or theorem, or just simple provable formulas. When building a proof system we choose not only axioms of the system, but also specific rules of inference. The choice of rules is often linked, as was the choi ...
MAT 300/371 Mathematical Structures/Advanced Calculus (Why) is
... (Why) is there a number whose square in 2 ? Theorem: The square of every rational number is different from 2. To check your proof, successively replace 2 by 3, 4, 6 and 180 and see if the argument still goes through. Recall the Bernoulli inequality: For h > −1 and n ∈ Z+ , (1 + h)n ≥ 1 + nh (easy pr ...
... (Why) is there a number whose square in 2 ? Theorem: The square of every rational number is different from 2. To check your proof, successively replace 2 by 3, 4, 6 and 180 and see if the argument still goes through. Recall the Bernoulli inequality: For h > −1 and n ∈ Z+ , (1 + h)n ≥ 1 + nh (easy pr ...
Propositional logic - Cheriton School of Computer Science
... Note that although we applied our rules to atoms in the previous proof, the rules are stated in terms of general formulas. In fact, because the rules talk about the “top-level” structure of the formulas to which they are applied, we can get other proofs from this proof almost for free. We can take ...
... Note that although we applied our rules to atoms in the previous proof, the rules are stated in terms of general formulas. In fact, because the rules talk about the “top-level” structure of the formulas to which they are applied, we can get other proofs from this proof almost for free. We can take ...
Section 1
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse a ...
... Contrapositives, converses, and inverses Definition Consider the implication p q 1. The converse of the implication is 2. The inverse of the implication is 3. The contrapositive of the implication is Proposition 3 1. An implication and its contrapositive are logically equivalent 2. The converse a ...
Euler`s totient function and Euler`s theorem
... (ii) any two distinct members are incongruent modulo n is called a reduced system of residues modulo n. Example 2.3. The canonical way to think of a reduced system of residues modulo an integer n ≥ 1 is by taking representatives from each of the ϕ(n) congruences classes identified by integers in [1, ...
... (ii) any two distinct members are incongruent modulo n is called a reduced system of residues modulo n. Example 2.3. The canonical way to think of a reduced system of residues modulo an integer n ≥ 1 is by taking representatives from each of the ϕ(n) congruences classes identified by integers in [1, ...
PDF
... In this entry, we show that the deduction theorem below holds for intuitionistic propositional logic. We use the axiom system provided in this entry. Theorem 1. If ∆, A `i B, where ∆ is a set of wff ’s of the intuitionistic propositional logic, then ∆ `i A → B. The proof is very similar to that of t ...
... In this entry, we show that the deduction theorem below holds for intuitionistic propositional logic. We use the axiom system provided in this entry. Theorem 1. If ∆, A `i B, where ∆ is a set of wff ’s of the intuitionistic propositional logic, then ∆ `i A → B. The proof is very similar to that of t ...
Lacunary recurrences for Eisenstein series
... University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative. 1Note that for k = 2, we have to fix a certain order of summation to ensure convergence of the defining double series a ...
... University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative. 1Note that for k = 2, we have to fix a certain order of summation to ensure convergence of the defining double series a ...
On a Symposium on the Foundations of Mathematics (1971) Paul
... axioms, as was done simultaneously by A.A. Fraenkel and Thoralf Skolem. The systems presented by W.V. Quine mediated, as it were, between type theory and the systems of axiomatic set theory. It turned out, however, by the results of Kurt Gödel and Skolem, that all such strictly formal frameworks fo ...
... axioms, as was done simultaneously by A.A. Fraenkel and Thoralf Skolem. The systems presented by W.V. Quine mediated, as it were, between type theory and the systems of axiomatic set theory. It turned out, however, by the results of Kurt Gödel and Skolem, that all such strictly formal frameworks fo ...
THE PRIME NUMBER THEOREM AND THE
... Every positive integer (other than 1) is either a prime or a product of primes. PROOF: Let Let n ≥ 2 be an any integer. If n is a prime, there is nothing to prove. If n is not a prime it has a divisor different from itself and from 1. If m is the smallest such divisor, then m must be a prime. Let’s ...
... Every positive integer (other than 1) is either a prime or a product of primes. PROOF: Let Let n ≥ 2 be an any integer. If n is a prime, there is nothing to prove. If n is not a prime it has a divisor different from itself and from 1. If m is the smallest such divisor, then m must be a prime. Let’s ...
Propositional Statements Direct Proof
... a and b are said to have the same parity if both are even or both are odd. Proof: Given x, we have 2 options in life and x must be either odd or even. Case I: x is odd, so x = 2n + 1 for some n ∈ Z. We want x + 204 = 2m + 1 for some m ∈ Z x + 204 = 2n + 1 + 204 = 2n + 205 = 2(n + 102) + 1 ...
... a and b are said to have the same parity if both are even or both are odd. Proof: Given x, we have 2 options in life and x must be either odd or even. Case I: x is odd, so x = 2n + 1 for some n ∈ Z. We want x + 204 = 2m + 1 for some m ∈ Z x + 204 = 2n + 1 + 204 = 2n + 205 = 2(n + 102) + 1 ...
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.