slides03 - Duke University
... • Now, just from this, what do we know about x and y? You should think back to the definition of rational: • … Since x is rational, we know (from the very definition of rational) that there must be some integers i and j such that x = i/j. So, let ix,jx be such integers … • We give them unique names ...
... • Now, just from this, what do we know about x and y? You should think back to the definition of rational: • … Since x is rational, we know (from the very definition of rational) that there must be some integers i and j such that x = i/j. So, let ix,jx be such integers … • We give them unique names ...
a simple derivation of jacobi`s four-square formula
... briefly here record that the theorem was conjectured by Bachet in 1621, was claimed to have been proved by Fermât, but was not actually proved until Lagrange did so in 1770. It should also be mentioned that Lagrange was greatly assisted by Euler, who derived an identity which was crucial in Lagrange ...
... briefly here record that the theorem was conjectured by Bachet in 1621, was claimed to have been proved by Fermât, but was not actually proved until Lagrange did so in 1770. It should also be mentioned that Lagrange was greatly assisted by Euler, who derived an identity which was crucial in Lagrange ...
I. Precisely complete the following definitions: 1. A natural number n
... 1. The power you want is the minimum of the numbers m and n. Prove this power of p divides a + b by a direct argument. Show no larger power of p divides a + b by contradiction. 2. See the proof by contradiction for the theorem proved in class: there are infinitely many primes of the form 4k + 3. The ...
... 1. The power you want is the minimum of the numbers m and n. Prove this power of p divides a + b by a direct argument. Show no larger power of p divides a + b by contradiction. 2. See the proof by contradiction for the theorem proved in class: there are infinitely many primes of the form 4k + 3. The ...
Löwenheim-Skolem theorems and Choice principles
... Theorem 1. LS(ℵ0 ): Every model M of a first order theory T with countable signature has an elementary submodel N which is at most countable. As noted by Skolem himself, the condition that the countable model for T be an elementary submodel of M made an apparently essential use of the axiom of choic ...
... Theorem 1. LS(ℵ0 ): Every model M of a first order theory T with countable signature has an elementary submodel N which is at most countable. As noted by Skolem himself, the condition that the countable model for T be an elementary submodel of M made an apparently essential use of the axiom of choic ...
Reasoning with Quantifiers
... That is, we assume that x is such that P(x) is true, and show that Q(x) must also be true. technique is called direct proof or generalizing from the generic particular. Example: Prove that the sum of any two even integers is even. That is, prove: ...
... That is, we assume that x is such that P(x) is true, and show that Q(x) must also be true. technique is called direct proof or generalizing from the generic particular. Example: Prove that the sum of any two even integers is even. That is, prove: ...
Discrete Mathematics I Lectures Chapter 4
... If the negation is TRUE, then the original statement must be FALSE!!! ...
... If the negation is TRUE, then the original statement must be FALSE!!! ...
3. CATALAN NUMBERS Corollary 1. cn = 1
... a0 ; a1 ; : : : ; an counterclockwise (the unmarked side is between a0 and an ). Then for every triangle of the triangulation repeatedly do the following. If one side of the triangle is empty (unmarked) and the two remaining sides are marked by the expressions p and q (looking counterclockwise from ...
... a0 ; a1 ; : : : ; an counterclockwise (the unmarked side is between a0 and an ). Then for every triangle of the triangulation repeatedly do the following. If one side of the triangle is empty (unmarked) and the two remaining sides are marked by the expressions p and q (looking counterclockwise from ...
Answers
... (d) True - a square is a special type of rectangle. However, the converse is false, not every rectangle is a square. (e) False (the circumference of a circle is only approximately 3 times the diameter) (f) True ...
... (d) True - a square is a special type of rectangle. However, the converse is false, not every rectangle is a square. (e) False (the circumference of a circle is only approximately 3 times the diameter) (f) True ...
Notes
... This can be shown in a strong sense as our examples suggest. We’ll examine this below. Do we know that any specification we could write down in mathematics or logic can be expressed as an OCaml SL specification? What about this “true” statement in mathematics? ∀u : term where type u = unit. ∃n : N. ...
... This can be shown in a strong sense as our examples suggest. We’ll examine this below. Do we know that any specification we could write down in mathematics or logic can be expressed as an OCaml SL specification? What about this “true” statement in mathematics? ∀u : term where type u = unit. ∃n : N. ...
A NOTE ON AN ADDITIVE PROPERTY OF PRIMES 1. Introduction
... Arithmetic”(FTA) witch states that each natural number n > 1 can be written in an essentially unique way as a product of distinct primes factors. But what is the behavior of primes in the “ additive context”? It is easy to check that a theorem like FTA can not be true. In fact all the integers of th ...
... Arithmetic”(FTA) witch states that each natural number n > 1 can be written in an essentially unique way as a product of distinct primes factors. But what is the behavior of primes in the “ additive context”? It is easy to check that a theorem like FTA can not be true. In fact all the integers of th ...
A Paedagogic Example of Cut-Elimination
... Obviously, by the reflexivity of the partial order, the axioms are true in every lattice. Since the meet and join of two lattice elements x and y are lower and upper bounds of {x, y}, respectively, x ∩ y is a lower bound of {x} and x ∪ y an upper bound of {y}. By transitivity, the soundness of ∩:lef ...
... Obviously, by the reflexivity of the partial order, the axioms are true in every lattice. Since the meet and join of two lattice elements x and y are lower and upper bounds of {x, y}, respectively, x ∩ y is a lower bound of {x} and x ∪ y an upper bound of {y}. By transitivity, the soundness of ∩:lef ...
PDF
... 20 . If r = 0, then we’re done, we can simply set a = 20 and b = 20(m − 1) (or vice versa if preferred), thanks to the theorem on multiples of abundant numbers. The other nine possible values of r are almost as easy to dispose of: If r = 2, then assign a = 20(m − 2) and b = 42. This works for m > 2. ...
... 20 . If r = 0, then we’re done, we can simply set a = 20 and b = 20(m − 1) (or vice versa if preferred), thanks to the theorem on multiples of abundant numbers. The other nine possible values of r are almost as easy to dispose of: If r = 2, then assign a = 20(m − 2) and b = 42. This works for m > 2. ...
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.