An Introduction to Discrete Mathematics: how to
... Mathematics, like swimming or basket weaving, is an activity that requires skill. To acqure proficiency, one must develop skill though active exercise. Passively reading, listening and watching would not suffice if one wished to become good at swimming or basket weaving, nor does it suffice if one w ...
... Mathematics, like swimming or basket weaving, is an activity that requires skill. To acqure proficiency, one must develop skill though active exercise. Passively reading, listening and watching would not suffice if one wished to become good at swimming or basket weaving, nor does it suffice if one w ...
ON THE DIVISIBILITY OF THE CLASS NUMBER OF
... extensions of the rationale each with class number divisible by 2. In facf, if the discriminant of the field contains more than two prime factors, then 2 divides the class number. Max Gut [ l ] generalized this result to show that there exist infinitely many quadratic imaginary fields each with clas ...
... extensions of the rationale each with class number divisible by 2. In facf, if the discriminant of the field contains more than two prime factors, then 2 divides the class number. Max Gut [ l ] generalized this result to show that there exist infinitely many quadratic imaginary fields each with clas ...
paper by David Pierce
... We make no new row for x4 , since 4 = 1 in Z/3Z, so x4 has already been defined. If we did try to make a row for x4 , using (2.4), then it would not agree with the row for x1 . Thus, although we can use equations (2.3) and (2.4) to give a definition, in Peano’s sense, of exponentiation in Z/3Z, those ...
... We make no new row for x4 , since 4 = 1 in Z/3Z, so x4 has already been defined. If we did try to make a row for x4 , using (2.4), then it would not agree with the row for x1 . Thus, although we can use equations (2.3) and (2.4) to give a definition, in Peano’s sense, of exponentiation in Z/3Z, those ...
UNIQUE FACTORIZATION IN MULTIPLICATIVE SYSTEMS
... Pu P2, " • • 1 Ph- Thus the set ilf can be characterized as all positive integers relatively prime to ph+i, • • • , Pr, if such primes exist. So the set ilf can be described in terms of the modulus ph+iph+2 • ■ • pr, which is less than n since h^l. This contradicts our basic hypothesis that n is the ...
... Pu P2, " • • 1 Ph- Thus the set ilf can be characterized as all positive integers relatively prime to ph+i, • • • , Pr, if such primes exist. So the set ilf can be described in terms of the modulus ph+iph+2 • ■ • pr, which is less than n since h^l. This contradicts our basic hypothesis that n is the ...
Elements of Modal Logic - University of Victoria
... α1 . . . αn β A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of the αi ’s are in Σ. Each system S determines a logic L(S), which is defined as the smallest set containing A that is closed under the rules of R. The logic pc has an associated system. Let Spc = (Apc , Rpc ) ...
... α1 . . . αn β A set Σ is said to be closed under an inference rule iff β ∈ Σ whenever all of the αi ’s are in Σ. Each system S determines a logic L(S), which is defined as the smallest set containing A that is closed under the rules of R. The logic pc has an associated system. Let Spc = (Apc , Rpc ) ...
PPT
... Def: For any real number x, the floor of x, written x, is the unique integer n such that n x < n + 1. It is the largest integer not exceeding x ( x). Def: For any real number x, the ceiling of x, written x, is the unique integer n such that n – 1 < x n. What is n? If k is an integer, what a ...
... Def: For any real number x, the floor of x, written x, is the unique integer n such that n x < n + 1. It is the largest integer not exceeding x ( x). Def: For any real number x, the ceiling of x, written x, is the unique integer n such that n – 1 < x n. What is n? If k is an integer, what a ...
LAWS OF LARGE NUMBERS FOR PRODUCT OF RANDOM
... About asymptotic gaussianity, the following result is stated and its proof is obtained by using the central limit theorem for martingales, Duhamel’s equation and properties of product integral (see [1]). Theorem 6 The process MSn converges on the Skorokhod function space D(0, T ) to −S ·M, ...
... About asymptotic gaussianity, the following result is stated and its proof is obtained by using the central limit theorem for martingales, Duhamel’s equation and properties of product integral (see [1]). Theorem 6 The process MSn converges on the Skorokhod function space D(0, T ) to −S ·M, ...
Two Irrational Numbers That Give the Last Non
... (Again, calculations were performed by Maple.) Despite this striking similarity between P and N , it turns out that P , like F , is irrational: Theorem 2. Let P = 0.d1 d2 d3 . . . dn . . . be the infinite decimal such that each digit dn = lnzd(nn ). Then, P is irrational. Before we begin with the (s ...
... (Again, calculations were performed by Maple.) Despite this striking similarity between P and N , it turns out that P , like F , is irrational: Theorem 2. Let P = 0.d1 d2 d3 . . . dn . . . be the infinite decimal such that each digit dn = lnzd(nn ). Then, P is irrational. Before we begin with the (s ...
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.