Mathematical Reasoning: Writing and Proof
... course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are to help students: Develop logical thinking skills and to develop the ability to think mo ...
... course in the college mathematics curriculum that introduces students to the processes of constructing and writing proofs and focuses on the formal development of mathematics. The primary goals of the text are to help students: Develop logical thinking skills and to develop the ability to think mo ...
2.1 inductive reasoning and conjecture ink.notebook
... • To show that a conjecture is true, you must show that it is true for all cases. • To show that a conjecture is false, you must find one counterexample. • A counterexample is a specific case for which the conjecture is false. ...
... • To show that a conjecture is true, you must show that it is true for all cases. • To show that a conjecture is false, you must find one counterexample. • A counterexample is a specific case for which the conjecture is false. ...
CONSECUTIVE EVEN NUMBER FINDING GRAPH (CENFG
... greater than 2 can be expressed as sum of two primes the following definitions and terminologies have been cited to understand the use of graph for Gold Bach conjecture. Definition 1. A graph G = (V, E) consists of a set of objects V= {V1 , V2 , V3 . . . }called vertices, and another set E = {e1 , e ...
... greater than 2 can be expressed as sum of two primes the following definitions and terminologies have been cited to understand the use of graph for Gold Bach conjecture. Definition 1. A graph G = (V, E) consists of a set of objects V= {V1 , V2 , V3 . . . }called vertices, and another set E = {e1 , e ...
b - FSU Computer Science
... An integer a is a quadratic residue with respect to n if: a is relatively prime to n and • there exists an integer b such that: a = b2 mod n ...
... An integer a is a quadratic residue with respect to n if: a is relatively prime to n and • there exists an integer b such that: a = b2 mod n ...
Week 4 – Complex Numbers
... π. Euler was a giant in 18th century mathematics and the most prolific mathematician ever. His most important contributions were in analysis (eg. on infinite series, calculus of variations). The study of topology arguably dates back to his solution of the Königsberg Bridge Problem. (Many books, part ...
... π. Euler was a giant in 18th century mathematics and the most prolific mathematician ever. His most important contributions were in analysis (eg. on infinite series, calculus of variations). The study of topology arguably dates back to his solution of the Königsberg Bridge Problem. (Many books, part ...
Mathematics of the Golden Section: from Euclid to contemporary
... period of the development of the Greek mathematics and created a strong base to the further development of mathematics. During more than two millenia “The Elements” remained a basic work on the “Elementary Mathematics”, which gave the origin of many fundamental theories of mathematics, in particular ...
... period of the development of the Greek mathematics and created a strong base to the further development of mathematics. During more than two millenia “The Elements” remained a basic work on the “Elementary Mathematics”, which gave the origin of many fundamental theories of mathematics, in particular ...
SOME INFINITE CLASSES OF FULLERENE GRAPHS 1 Introduction
... edge of the hexagon to make all 6 neighboring pentagons, hexagon, and add an edge to each new vertex and finally join the ends of new edges to make a new hexagon. With this process, we get a new fullerene with 6 more hexagons. The new fullerene has the same property and we may do the process again t ...
... edge of the hexagon to make all 6 neighboring pentagons, hexagon, and add an edge to each new vertex and finally join the ends of new edges to make a new hexagon. With this process, we get a new fullerene with 6 more hexagons. The new fullerene has the same property and we may do the process again t ...
Divisibility and Congruence Definition. Let a ∈ Z − {0} and b ∈ Z
... 11. Cool Property 1. For all integers a, b, x, y, if a ≡ x mod m and b ≡ y mod m, then a + b ≡ x + y mod m. Proof. By hypothesis m | (x − a) and m | (y − b). This means (x − a) = mp and (y − b) = mq for some p, q ∈ Z. Adding gives ((x + y) − (a + b)) = (x − a) + (y − b) = m(p + q), and so m | (x + ...
... 11. Cool Property 1. For all integers a, b, x, y, if a ≡ x mod m and b ≡ y mod m, then a + b ≡ x + y mod m. Proof. By hypothesis m | (x − a) and m | (y − b). This means (x − a) = mp and (y − b) = mq for some p, q ∈ Z. Adding gives ((x + y) − (a + b)) = (x − a) + (y − b) = m(p + q), and so m | (x + ...
On the independence numbers of the powers of graph
... Calculating the Shannon capacity (motivated by Information Theory) is considered very difficult and the problem remains open even for such a simple graph as C7 . The best known upper bounds on the Shannon capacities of graphs are given by the Lovasz theta function [2]. The upper bound suffices to es ...
... Calculating the Shannon capacity (motivated by Information Theory) is considered very difficult and the problem remains open even for such a simple graph as C7 . The best known upper bounds on the Shannon capacities of graphs are given by the Lovasz theta function [2]. The upper bound suffices to es ...
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.