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Section 1.4 Mathematical Proofs
Section 1.4 Mathematical Proofs

The theorem, it`s meaning and the central concepts
The theorem, it`s meaning and the central concepts

Fractions don`t exist
Fractions don`t exist

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... σ 2 , then for any a > 0, σ2 P(|X − µ| ≥ a) ≤ 2 . a Example 3 Suppose that it is known that the number of items produced in a factory during a week is a random variable with mean 50. (a) What can be said about the probability that this week’s production will exceed 75? (b) If the variance of a weeks ...
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... Compare rational and irrational numbers to demonstrate that the decimal expansion of irrational numbers do not repeat; show that every rational number has a decimal expansion which eventually repeats and covert such decimals into rational numbers. Use rational numbers to approximate and locate irrat ...
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... admits no (strictly) positive integer solution. (this theorem, known as Fermat’s Last Theorem has been proved by Andrew Wiles in 1994). 2. Proofs by contradiction - the method The example we saw in previous section is called a direct proof. We will see a useful tool when direct proofs do not work. T ...
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creating mathematical knowledge
creating mathematical knowledge

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Inequalities in 2 triangles and indirect proofs

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PROOFS Math 174 May 2017 I. Introduction. In the natural sciences

Proof Example: The Irrationality of √ 2 During the lecture a student
Proof Example: The Irrationality of √ 2 During the lecture a student

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... that groups of B*A%' s repeat only if 5 = BE9 we will be done. By renaming "Anu to be "AQ9" we may use the result given in Theorem 2 to describe the positions of the S's in Am. We will show that for all integers i J> 1, there exist Oi z S and di i S such that
Homework 1 (Due Tuesday April 5)
Homework 1 (Due Tuesday April 5)

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Full text

... The aim of this paper is to establish an asymptotic formula for the summatory function of
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deductive system

Solutions
Solutions

Document
Document

Document
Document

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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.
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