Direct Proof
... Given integers a and d with d > 0, there exist unique integers q and r for which a = dq + r and 0 ≤ r < d. We call d the divisor, q is the quotient, and r is the remainder. Note: The remainder r is zero if d|a, otherwise it is a positive number strictly less than the divisor d. If the division algor ...
... Given integers a and d with d > 0, there exist unique integers q and r for which a = dq + r and 0 ≤ r < d. We call d the divisor, q is the quotient, and r is the remainder. Note: The remainder r is zero if d|a, otherwise it is a positive number strictly less than the divisor d. If the division algor ...
←→ ↓ ↓ ←→ ←→ ←→ ←→ −→ −→ → The diagonal lemma as
... Notation Our formal language is that of first-order arithmetic; F m is the set of formulas with at most one free variable; g is any one of the standard Gödel numberings; N denotes the set of Gödel numbers of formulas in F m; Q stands for Robinson arithmetic. The closed terms corresponding to natur ...
... Notation Our formal language is that of first-order arithmetic; F m is the set of formulas with at most one free variable; g is any one of the standard Gödel numberings; N denotes the set of Gödel numbers of formulas in F m; Q stands for Robinson arithmetic. The closed terms corresponding to natur ...
The Anti-Foundation Axiom in Constructive Set Theories
... Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-well ...
... Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-well ...
CE221_week_1_Chapter1_Introduction
... • It proceeds by assuming that the theorem is false and showing that this assumption implies that some known property is false, and hence the original assumption is erroneous. • Example: Prove that there is an infinite number of primes. • Proof: Assume the theorem is false, so that there is some lar ...
... • It proceeds by assuming that the theorem is false and showing that this assumption implies that some known property is false, and hence the original assumption is erroneous. • Example: Prove that there is an infinite number of primes. • Proof: Assume the theorem is false, so that there is some lar ...
Full text
... 2: Let IT be a partition whose Ferrers graph is embedded in the fourth quadrant. Each node (i, j) of the fourth quadrant which is not in the Ferrers graph of IT is said to possess an anti-hook difference p^ - kj relative to ir, where p^ is the number of nodes on the i t h row of the fourth quadrant ...
... 2: Let IT be a partition whose Ferrers graph is embedded in the fourth quadrant. Each node (i, j) of the fourth quadrant which is not in the Ferrers graph of IT is said to possess an anti-hook difference p^ - kj relative to ir, where p^ is the number of nodes on the i t h row of the fourth quadrant ...
CSE 20 - Lecture 14: Logic and Proof Techniques
... Negating a sentence: iclicker What is the negation of the sentence: “There is an university in USA where every department has at least 20 faculty and at least one noble laureate.” There is an university in USA where every department has less than 20 faculty and at least one noble laureate. All univ ...
... Negating a sentence: iclicker What is the negation of the sentence: “There is an university in USA where every department has at least 20 faculty and at least one noble laureate.” There is an university in USA where every department has less than 20 faculty and at least one noble laureate. All univ ...
Vector length bound
... Claim 1: ai+1 || bi for all i (with wrap-around) Proof: Since each proc. does all sends before any receives, there is no transitivity. Also pi+1 does not send to pi. Claim 2: ai+1 bj for all j ≠ i. Proof: If j = i+1, obvious. If j ≠ i+1, then pi+1 sends to pj: ...
... Claim 1: ai+1 || bi for all i (with wrap-around) Proof: Since each proc. does all sends before any receives, there is no transitivity. Also pi+1 does not send to pi. Claim 2: ai+1 bj for all j ≠ i. Proof: If j = i+1, obvious. If j ≠ i+1, then pi+1 sends to pj: ...
a proof for goldbach`s conjecture
... In 1937, Vinograd of Russian Mathematician proved that each odd large number can be shown by three primes. In 1930, Lev Schnirelmann proved that each natural number can be shown by M-primes. In 1973, Chen Jingrun proved that each odd number can be shown by one prime plus a number that has maximum tw ...
... In 1937, Vinograd of Russian Mathematician proved that each odd large number can be shown by three primes. In 1930, Lev Schnirelmann proved that each natural number can be shown by M-primes. In 1973, Chen Jingrun proved that each odd number can be shown by one prime plus a number that has maximum tw ...
Full text
... By (iii) of Theorem 2, each number in the n row is the sum of the three adjacent numbers in the (n - 1) and (n - 2)nd rows. For example, the number 25 in the 5 t h row is the sum of its three adjacent numbers 5, 7,13 in the 3 rc! and 4 t h rows. Therefore, instead of using the formula in Theorem 1, ...
... By (iii) of Theorem 2, each number in the n row is the sum of the three adjacent numbers in the (n - 1) and (n - 2)nd rows. For example, the number 25 in the 5 t h row is the sum of its three adjacent numbers 5, 7,13 in the 3 rc! and 4 t h rows. Therefore, instead of using the formula in Theorem 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.