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(1) Find all prime numbers smaller than 100. (2) Give a proof by
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
Algebra 2, with Trig
... When we find the roots of a polynomial equation, we are finding the places where the value of the function is ZERO. If f(x) = 0, then x is a root!!!! Now we have to connect this idea with synthetic substitution. Remember that synthetic substitution allows us to find the value of a function for a giv ...
... When we find the roots of a polynomial equation, we are finding the places where the value of the function is ZERO. If f(x) = 0, then x is a root!!!! Now we have to connect this idea with synthetic substitution. Remember that synthetic substitution allows us to find the value of a function for a giv ...
Lecture 3
... Superman Proof in CafeOBJ • Note in the following proof we do not need the truth values of the equations from the SUMPERMAN module. We only need the constants representing the sentences. The proof uses the given equations and general theorems of equational logic. The proof does not depend on the va ...
... Superman Proof in CafeOBJ • Note in the following proof we do not need the truth values of the equations from the SUMPERMAN module. We only need the constants representing the sentences. The proof uses the given equations and general theorems of equational logic. The proof does not depend on the va ...
3x9: 9 E 9}, V{ A 8: 9 ES)
... above, to contain just one element. Thus, by (i), |6„| < k, contradicting the largeness of a. Now, using Claim 8, one constructs precisely as in the proof of Theorem 1, a tree 7;, s E<"2 s.t. for all s: (a) Ts is finite, a E Ts and /\TS is large; (b) there is \p E ^ s.t. \L E TsA<0y and -jt// E rjA< ...
... above, to contain just one element. Thus, by (i), |6„| < k, contradicting the largeness of a. Now, using Claim 8, one constructs precisely as in the proof of Theorem 1, a tree 7;, s E<"2 s.t. for all s: (a) Ts is finite, a E Ts and /\TS is large; (b) there is \p E ^ s.t. \L E TsA<0y and -jt// E rjA< ...
THE FERMAT EQUATION 1. Fermat`s Last Theorem for n = 4 The proof
... 6th root of unity. It turns out that we are lucky that the ring of all algebraic integers in Q[ζ6 ] – which itself turns out to be Z[ζ6 ] = {a + bζ6 | a, b ∈ Z} – is a Principal Ideal Domain. The proof is similar to that for the Gaussian integers – we show that any element of the quotient field Q[ζ6 ...
... 6th root of unity. It turns out that we are lucky that the ring of all algebraic integers in Q[ζ6 ] – which itself turns out to be Z[ζ6 ] = {a + bζ6 | a, b ∈ Z} – is a Principal Ideal Domain. The proof is similar to that for the Gaussian integers – we show that any element of the quotient field Q[ζ6 ...
A product of Gamma function values at fractions with the
... A BSTRACT. We give an exact formula for the product of the values of Euler’s Gamma function evaluated at all rational numbers between 0 and 1 with the same denominator in lowest terms; the answer depends on whether or not that denominator is a prime power. A consequence is a surprisingly nice formul ...
... A BSTRACT. We give an exact formula for the product of the values of Euler’s Gamma function evaluated at all rational numbers between 0 and 1 with the same denominator in lowest terms; the answer depends on whether or not that denominator is a prime power. A consequence is a surprisingly nice formul ...
The pigeonhole principle
... Regular expressions over Σ: [Basis] 1. Φ and σ for each σ Σ are regular expressions. [Inductive step] If α and β are regular expressions, then so are: ...
... Regular expressions over Σ: [Basis] 1. Φ and σ for each σ Σ are regular expressions. [Inductive step] If α and β are regular expressions, then so are: ...
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.