Full text
... (Without some gloss, this last statement is false; for example, 7 divides 49 2 + 1962. No doubt y and z are supposed to be mutually prime.) If his paper had been written by a much less eminent mathematician, I might have suspected that his claims were based in part on numerical evidence and not on c ...
... (Without some gloss, this last statement is false; for example, 7 divides 49 2 + 1962. No doubt y and z are supposed to be mutually prime.) If his paper had been written by a much less eminent mathematician, I might have suspected that his claims were based in part on numerical evidence and not on c ...
Lecture 3 - CSE@IIT Delhi
... Translating Mathematical Theorem Goldbach’s conjecture: Every even number is the sum of two prime numbers. ...
... Translating Mathematical Theorem Goldbach’s conjecture: Every even number is the sum of two prime numbers. ...
Infinite Games - International Mathematical Union
... 5.3 LEMMA. If II has a winning strategy for G*, then II has a winning strategy for G. Let s* be a winning strategy for II for G*. We define a strategy s for II for G. Let o- b e a position in G with II to move. Let ß09 ...,ßm be even ordinals such that, if G is extended to a position G* in G* ,by se ...
... 5.3 LEMMA. If II has a winning strategy for G*, then II has a winning strategy for G. Let s* be a winning strategy for II for G*. We define a strategy s for II for G. Let o- b e a position in G with II to move. Let ß09 ...,ßm be even ordinals such that, if G is extended to a position G* in G* ,by se ...
Newsletter – Ch 7
... 8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.G.6: Explain a proof of the Pyth ...
... 8.EE.2: Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 8.G.6: Explain a proof of the Pyth ...
1 Cardinality and the Pigeonhole Principle
... A set A is said to be finite if |A| = |Nn | for some n. In this case we say that |A| = n. Again, this is secretly a common sense definition, or at least one you’ve known since you were very young. When we want to know how many apples are in a bag and don’t have your standardized bags of oranges2 to ...
... A set A is said to be finite if |A| = |Nn | for some n. In this case we say that |A| = n. Again, this is secretly a common sense definition, or at least one you’ve known since you were very young. When we want to know how many apples are in a bag and don’t have your standardized bags of oranges2 to ...
On writing proofs
... The majority of proofs are implications – you are asked to prove a statement of the form: “If a, then b.” In symbols, “a ⇒ b.” (The symbol → is not a good substitute for the symbol ⇒.) Statement a is called the premise (or synonymously, assumption or hypothesis) and statement b is called the conclus ...
... The majority of proofs are implications – you are asked to prove a statement of the form: “If a, then b.” In symbols, “a ⇒ b.” (The symbol → is not a good substitute for the symbol ⇒.) Statement a is called the premise (or synonymously, assumption or hypothesis) and statement b is called the conclus ...
journal of number theory 13, 446
... We will defer the proofs of these results . Note that the import of Theorem 3 .2 is that the perturbations can take place at arbitrarily sparse points, given the stronger condition on S . Another difference between Theorems 3 .1 and 3 .2 is that in the latter, the perturbations can be required to be ...
... We will defer the proofs of these results . Note that the import of Theorem 3 .2 is that the perturbations can take place at arbitrarily sparse points, given the stronger condition on S . Another difference between Theorems 3 .1 and 3 .2 is that in the latter, the perturbations can be required to be ...
Conventions for Writing Mathematical Proofs
... • It’s bad form to begin a sentence with a mathematical symbol, because it makes it hard for the reader to recognize that a new sentence has begun. (You can’t capitalize a symbol to indicate the beginning of a sentence!) It’s usually easy to avoid this by minor rewording—for example, if you find you ...
... • It’s bad form to begin a sentence with a mathematical symbol, because it makes it hard for the reader to recognize that a new sentence has begun. (You can’t capitalize a symbol to indicate the beginning of a sentence!) It’s usually easy to avoid this by minor rewording—for example, if you find you ...
A Brief Introduction to the Intuitionistic Propositional Calculus
... intuitionism in very different ways. The problem here is not just that it means that you can only hope to pin them down one at a time (although it does indeed mean that), but the very fact that the fact that so many different ways of thinking lead to the same theory is a powerful argument for the na ...
... intuitionism in very different ways. The problem here is not just that it means that you can only hope to pin them down one at a time (although it does indeed mean that), but the very fact that the fact that so many different ways of thinking lead to the same theory is a powerful argument for the na ...
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.