Solutions to selected homework problems
... which is divisible by 5 since n5 − n is divisible by 5 (by induction hypothesis). Problem: Show that every nonzero integer can be uniquely represented as: ek 3k + ek−1 3k−1 + · · · + e1 3 + e0 , where ej = −1, 0, 1 and ek 6= 0. Solution: To prove that any number can be represented this way just mimi ...
... which is divisible by 5 since n5 − n is divisible by 5 (by induction hypothesis). Problem: Show that every nonzero integer can be uniquely represented as: ek 3k + ek−1 3k−1 + · · · + e1 3 + e0 , where ej = −1, 0, 1 and ek 6= 0. Solution: To prove that any number can be represented this way just mimi ...
3.4 Complex Zeros and the Fundamental Theorem of
... takes time to digest. Don’t be overly concerned if it doesn’t seem to sink in all at once, and pace yourself in the Exercises or you’re liable to get mental cramps. But before we get to the Exercises, we’d like to offer a bit of an epilogue. Our main goal in presenting the material on the complex ze ...
... takes time to digest. Don’t be overly concerned if it doesn’t seem to sink in all at once, and pace yourself in the Exercises or you’re liable to get mental cramps. But before we get to the Exercises, we’d like to offer a bit of an epilogue. Our main goal in presenting the material on the complex ze ...
ON A LEMMA OF LITTLEWOOD AND OFFORD
... 2^J»i€^jb and ]Cfc=i*& #& are both in 7, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less than 2. Now a theorem of Sperner 2 states that in any collection of subsets of n elements such that of every pair of subsets neither contains ...
... 2^J»i€^jb and ]Cfc=i*& #& are both in 7, neither of the corresponding subsets can contain the other, for otherwise their difference would clearly be not less than 2. Now a theorem of Sperner 2 states that in any collection of subsets of n elements such that of every pair of subsets neither contains ...
Relative normalization
... proof-language of T is complex: it contains proof-variables, proof-terms, as well as the terms of the theory T (that appear in proof-terms). Moreover, we need to express usual syntactic operations, such as α-conversion, substitution, etc. For that let us consider a language of trees L generated by a ...
... proof-language of T is complex: it contains proof-variables, proof-terms, as well as the terms of the theory T (that appear in proof-terms). Moreover, we need to express usual syntactic operations, such as α-conversion, substitution, etc. For that let us consider a language of trees L generated by a ...
Methods of Proof Ch 11
... even though no one since was able to do so until 1995 when Andrew Wiles of Princeton produced a fantastic 200 page proof ending 350 years of intense study.\ In general, a statement is not considered to be a theorem simply because it is true. A theorem carries a mathematical significance by virtue of ...
... even though no one since was able to do so until 1995 when Andrew Wiles of Princeton produced a fantastic 200 page proof ending 350 years of intense study.\ In general, a statement is not considered to be a theorem simply because it is true. A theorem carries a mathematical significance by virtue of ...
Pythagorean Theorem
... Does it matter whether we use a = 8 or 15? No. Let’s use a = 8 and b = 15. ...
... Does it matter whether we use a = 8 or 15? No. Let’s use a = 8 and b = 15. ...
A Tour of Formal Verification with Coq:Knuth`s Algorithm for Prime
... There is no relation between the length of a program and the difficulty of its proof of correctness. Very long programs performing elementary tasks could be trivial to prove correct, while short programs relying on some very deep properties could be much harder. Highly optimized programs usually bel ...
... There is no relation between the length of a program and the difficulty of its proof of correctness. Very long programs performing elementary tasks could be trivial to prove correct, while short programs relying on some very deep properties could be much harder. Highly optimized programs usually bel ...
The number of rational numbers determined by large sets of integers
... When A and B are subsets of the positive integers let A/B be the set of all rational numbers expressible as a/b with (a, b) in A × B. Suppose now that A and B are intervals in the integers in [1, X] and [1, Y ] respectively, satisfying |A| αX and |B| βY , where X, Y real numbers at least 1, α, β ...
... When A and B are subsets of the positive integers let A/B be the set of all rational numbers expressible as a/b with (a, b) in A × B. Suppose now that A and B are intervals in the integers in [1, X] and [1, Y ] respectively, satisfying |A| αX and |B| βY , where X, Y real numbers at least 1, α, β ...
Lecture 9: Integers, Rational Numbers and Algebraic Numbers
... We can without loss of generality assume that p and q have no common divisors (i.e., that the fraction pq is reduced as far as possible). We have 2q 2 = p2 so p2 is even. Hence p is even. Therefore, p is of the form p = 2k for some k ∈ Z. But then 2q2 = 4k2 or q 2 = 2k 2 so q is even, so p and q hav ...
... We can without loss of generality assume that p and q have no common divisors (i.e., that the fraction pq is reduced as far as possible). We have 2q 2 = p2 so p2 is even. Hence p is even. Therefore, p is of the form p = 2k for some k ∈ Z. But then 2q2 = 4k2 or q 2 = 2k 2 so q is even, so p and q hav ...
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.