A Note on Naive Set Theory in LP
... natural paraconsistent expansion of classical predicate logic. It leaves all things in predicate logic as they are, except to allow that sentences could be both true and false. In particular, in any consistent fragment of its domain, LP acts identically to the classical predicate calculus. The resul ...
... natural paraconsistent expansion of classical predicate logic. It leaves all things in predicate logic as they are, except to allow that sentences could be both true and false. In particular, in any consistent fragment of its domain, LP acts identically to the classical predicate calculus. The resul ...
Fermat*s Little Theorem (2/24)
... Suppose n is some odd number and we’d like to know if it’s composite, but we’re having trouble factoring it. Well, compute 2n –1 (mod n). What if the answer is not 1? Example. I wonder if 376289 is prime? Using a computer, I find that 2376288 150132 (mod 376289). Conclusion? In fact 376289 = 571 659 ...
... Suppose n is some odd number and we’d like to know if it’s composite, but we’re having trouble factoring it. Well, compute 2n –1 (mod n). What if the answer is not 1? Example. I wonder if 376289 is prime? Using a computer, I find that 2376288 150132 (mod 376289). Conclusion? In fact 376289 = 571 659 ...
The Euler characteristic of the moduli space of curves
... H 4i+ 2(Sp; (~) [2"]) suggests that the contribution to the large Euler characteristic from the stable part of the cohomology may be relatively small. The formula for Z(Fg1) will follow from two other theorems, which we now state. For every positive integer n > 0 let ~, denote a fixed 2n-gon with it ...
... H 4i+ 2(Sp; (~) [2"]) suggests that the contribution to the large Euler characteristic from the stable part of the cohomology may be relatively small. The formula for Z(Fg1) will follow from two other theorems, which we now state. For every positive integer n > 0 let ~, denote a fixed 2n-gon with it ...
Contradiction: means to follow a path toward which a statement
... (¬q ^ p) → (r ^ ¬r), where (r ^ ¬r) isn’t even the conclusion to the original hypothesis it’s something else that’s a result of the proof Step 1) Assume the hypothesis and the negation of the conclusion to be true. Step 2) Perform operations such that a conclusion is reached which is not true For in ...
... (¬q ^ p) → (r ^ ¬r), where (r ^ ¬r) isn’t even the conclusion to the original hypothesis it’s something else that’s a result of the proof Step 1) Assume the hypothesis and the negation of the conclusion to be true. Step 2) Perform operations such that a conclusion is reached which is not true For in ...
constant curiosity - users.monash.edu.au
... let’s spare a thought for a few of the lesser known mathematical constants — ones which might not permeate the various fields of mathematics but have nevertheless been immortalised in the mathematical literature in one way or another. In this seminar, we’ll consider a few of these numerical curios a ...
... let’s spare a thought for a few of the lesser known mathematical constants — ones which might not permeate the various fields of mathematics but have nevertheless been immortalised in the mathematical literature in one way or another. In this seminar, we’ll consider a few of these numerical curios a ...
Lecture Notes - jan.ucc.nau.edu
... Proof by Contradiction: Example • Theorem: There exists an infinite number of prime numbers. • Proof (courtesy of Euclid): 1. Assume that there are a finite number of primes. 2. Then there is a largest prime, p. Consider the number q = (2x3x5x7x...xp)+1. q is one more than the product of all primes ...
... Proof by Contradiction: Example • Theorem: There exists an infinite number of prime numbers. • Proof (courtesy of Euclid): 1. Assume that there are a finite number of primes. 2. Then there is a largest prime, p. Consider the number q = (2x3x5x7x...xp)+1. q is one more than the product of all primes ...
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.