What is a proof? - Computer Science
... simplest form as follows: We have n + 1 pigeons and n holes. We put all the pigeons in holes (in any way we want). The principle tells us that there must be at least one hole with at least two pigeons in it. Why is that true? Try to visualize the example of n = 2; therefore, we have 3 pigeons and 2 ...
... simplest form as follows: We have n + 1 pigeons and n holes. We put all the pigeons in holes (in any way we want). The principle tells us that there must be at least one hole with at least two pigeons in it. Why is that true? Try to visualize the example of n = 2; therefore, we have 3 pigeons and 2 ...
8-5 Angle Bisector Theorem
... illustrated in the accompanying diagram. The angle between Poplar Street and Fern is equivalent to the angle between Elm Street and Fern street. If the distance from E to F is 8 miles, F to P is 10 miles, and other distances are indicated below, how far apart is are points ...
... illustrated in the accompanying diagram. The angle between Poplar Street and Fern is equivalent to the angle between Elm Street and Fern street. If the distance from E to F is 8 miles, F to P is 10 miles, and other distances are indicated below, how far apart is are points ...
IRRATIONALITY OF π AND e 1. Introduction Numerical estimates for
... are based on techniques from calculus, which can be used to prove irrationality of other numbers, such as e and rational powers of e (aside from e0 = 1). The remaining sections are organized as follows. In Section 2, we prove π is irrational with definite integrals. Irrationality of e is proved by i ...
... are based on techniques from calculus, which can be used to prove irrationality of other numbers, such as e and rational powers of e (aside from e0 = 1). The remaining sections are organized as follows. In Section 2, we prove π is irrational with definite integrals. Irrationality of e is proved by i ...
A Generalization of the Congruent Number Problem
... The problem of classifying congruent numbers reduces to the cases where n is square-free. We can scale areas trivially. We can easily generate examples of congruent numbers; for example 6 is congruent and is given by the 3 − 4 − 5 triangle. Classically, people were able to solve this problem in a fe ...
... The problem of classifying congruent numbers reduces to the cases where n is square-free. We can scale areas trivially. We can easily generate examples of congruent numbers; for example 6 is congruent and is given by the 3 − 4 − 5 triangle. Classically, people were able to solve this problem in a fe ...
Section 8.2 Markov and Chebyshev Inequalities and the Weak Law
... which is the general result. EXAMPLE: An astronomer is measuring the distance to a star. Because of different errors, each measurement will not be precisely correct, but merely an estimate. He will therefore make a series of measurements and use the average as his estimate of the distance. He beli ...
... which is the general result. EXAMPLE: An astronomer is measuring the distance to a star. Because of different errors, each measurement will not be precisely correct, but merely an estimate. He will therefore make a series of measurements and use the average as his estimate of the distance. He beli ...
Chapter 13
... algebra to analyze and describe the behavior of circuits built with relays. This adaptation is called switching algebra. ...
... algebra to analyze and describe the behavior of circuits built with relays. This adaptation is called switching algebra. ...
Lecture 12 - stony brook cs
... Main Factorization Theorem We are now ready to prove the main theorem about factorization. The idea of this theorem, as well as all Facts 1-5 we will use in proving it, can be found in Euclid’s Elements in Book VII and Book IX Main Factorization Theorem Every composite number can be factored unique ...
... Main Factorization Theorem We are now ready to prove the main theorem about factorization. The idea of this theorem, as well as all Facts 1-5 we will use in proving it, can be found in Euclid’s Elements in Book VII and Book IX Main Factorization Theorem Every composite number can be factored unique ...
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.