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Lecture 5: Ramsey Theory 1 Ramsey`s theorem for graphs
... a large system, however complicated, there is always a smaller subsystem which exhibits some sort of special structure. Perhaps the oldest statement of this type is the following. Proposition 1. Among any six people, there are three any two of whom are friends, or there are three such that no two of ...
... a large system, however complicated, there is always a smaller subsystem which exhibits some sort of special structure. Perhaps the oldest statement of this type is the following. Proposition 1. Among any six people, there are three any two of whom are friends, or there are three such that no two of ...
Erd˝os`s proof of Bertrand`s postulate
... The Riemann hypothesis would imply that we could shorten this interval to (n, n+n 2 + ], and a very strong conjecture of Cramér [2] would imply we could shorten it to (n, n+(1+) ln2 n]. Here is a very lovely open question much in the spirit of Bertrand’s postulate. Question 3.1 Is it true that fo ...
... The Riemann hypothesis would imply that we could shorten this interval to (n, n+n 2 + ], and a very strong conjecture of Cramér [2] would imply we could shorten it to (n, n+(1+) ln2 n]. Here is a very lovely open question much in the spirit of Bertrand’s postulate. Question 3.1 Is it true that fo ...
The Compactness Theorem for first-order logic
... Now lets enlarge our language L by adding a constant symbol δ. (We’ll think of δ as an infinitesimally small number). For each n, let φn be the sentence: φn = 0 < δ < 1/n Now consider the theory T 0 = T ∪ {φ1 , φ2 , . . .}. We claim that T 0 is satisfiable. This is by the compactness theorem. If T0 ...
... Now lets enlarge our language L by adding a constant symbol δ. (We’ll think of δ as an infinitesimally small number). For each n, let φn be the sentence: φn = 0 < δ < 1/n Now consider the theory T 0 = T ∪ {φ1 , φ2 , . . .}. We claim that T 0 is satisfiable. This is by the compactness theorem. If T0 ...
Oliver Johnson and Christina Goldschmidt 1. Introduction
... Theorem 1.4. If V and W are independent log-concave random variables, then their sum V + W is also log-concave. Equivalently, the convolution of any two log-concave sequences is log-concave. A similar result was proved by Davenport and Pólya [7], under the condition that the probability mass functi ...
... Theorem 1.4. If V and W are independent log-concave random variables, then their sum V + W is also log-concave. Equivalently, the convolution of any two log-concave sequences is log-concave. A similar result was proved by Davenport and Pólya [7], under the condition that the probability mass functi ...
21.3 Prime factors
... which states that every integer x > 1 can be written uniquely in the form x = pk11 pk22 · · · pkr r , where the pi ’s are primes and the ki ’s are positive integers. Given x, we are interested in the number r of prime factors of x, that is, in the number of distinct primes pi in such a representatio ...
... which states that every integer x > 1 can be written uniquely in the form x = pk11 pk22 · · · pkr r , where the pi ’s are primes and the ki ’s are positive integers. Given x, we are interested in the number r of prime factors of x, that is, in the number of distinct primes pi in such a representatio ...
Informal proofs
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • P ...
... Methods of proving theorems Basic methods to prove the theorems: • Direct proof – p q is proved by showing that if p is true then q follows • Indirect proof – Show the contrapositive ¬q ¬p. If ¬q holds then ¬p follows • Proof by contradiction – Show that (p ¬ q) contradicts the assumptions • P ...
Section 3.6: Indirect Argument: Contradiction and Contraposition
... So far, we have only considered so called “direct proofs” of mathematical statements. Specifically, we have been given a statement to prove, and then we have used the definitions and previous results to logically derive the statement. In this section we consider “indirect proofs” proofs which do not ...
... So far, we have only considered so called “direct proofs” of mathematical statements. Specifically, we have been given a statement to prove, and then we have used the definitions and previous results to logically derive the statement. In this section we consider “indirect proofs” proofs which do not ...
Practice Midterm 1
... (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. ...
... (b) Suppose that 3 was a positive rational number. Let us say that 3 is equal to c/d, where c and d are natural numbers. Use part 4(a) to show that 3d − c < c. ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.