Density of the Rationals and Irrationals in R
... Definition Let X be an ordered set. We say that Y ⊂ X is bounded from above if there exists x ∈ X such that x ≤ y for all x ∈ X, which we sometimes write as Y ≤ x. We say that X has the least upper bound property if for any Y ⊂ X which is bounded above, the collection {x ∈ X : Y ≤ x} of upper bounds ...
... Definition Let X be an ordered set. We say that Y ⊂ X is bounded from above if there exists x ∈ X such that x ≤ y for all x ∈ X, which we sometimes write as Y ≤ x. We say that X has the least upper bound property if for any Y ⊂ X which is bounded above, the collection {x ∈ X : Y ≤ x} of upper bounds ...
ordinals proof theory
... Theorem 2. If α is an element of ǫ0 , there exists a natural number k such that α < ω ↑ k. In [1], it is shown how to prove that ǫ0 is a well-order. This is obvious from the usual von Newman definition of ordinals. However, the proof in [1] is taken from the finitist point of view. Theorem 3. Every ...
... Theorem 2. If α is an element of ǫ0 , there exists a natural number k such that α < ω ↑ k. In [1], it is shown how to prove that ǫ0 is a well-order. This is obvious from the usual von Newman definition of ordinals. However, the proof in [1] is taken from the finitist point of view. Theorem 3. Every ...
Chapter 0 - Ravikumar - Sonoma State University
... • Assertions: Mathematical statement expresses some property of a set of defined objects. Assertions may or may not be true. ...
... • Assertions: Mathematical statement expresses some property of a set of defined objects. Assertions may or may not be true. ...
Week 13
... Perhaps the most famous theorem in all of mathematics is the Pythagorean Theorem. Dozens of very different proofs of the theorem have been devised, and many of them can be given in the form of “proofs without words.” The diagram to the right shows one of the most straightforward and accessible proof ...
... Perhaps the most famous theorem in all of mathematics is the Pythagorean Theorem. Dozens of very different proofs of the theorem have been devised, and many of them can be given in the form of “proofs without words.” The diagram to the right shows one of the most straightforward and accessible proof ...
Parry A
... Applies the distance and midpoint formulas 26. Right Triangles States and applies the Pythagorean Theorem and its converse. Warm up/Opener: Viewgraph 6.4 Essential Questions: What is special about right triangles? Activities: Lesson/Body: 6.5: We did what we need to do with congruence testing with r ...
... Applies the distance and midpoint formulas 26. Right Triangles States and applies the Pythagorean Theorem and its converse. Warm up/Opener: Viewgraph 6.4 Essential Questions: What is special about right triangles? Activities: Lesson/Body: 6.5: We did what we need to do with congruence testing with r ...
Math 2534 Test 1B Solutions
... prove the following Theorem: List of previously proved theorems: Refer to needed theorems only. Do not try to reprove any of these theorems. Two consecutive integers have opposite parity The product of two even (odd) integers is even (odd). The sum of any two odd integers or any two even integers is ...
... prove the following Theorem: List of previously proved theorems: Refer to needed theorems only. Do not try to reprove any of these theorems. Two consecutive integers have opposite parity The product of two even (odd) integers is even (odd). The sum of any two odd integers or any two even integers is ...
Determine the number of odd binomial coefficients in the expansion
... binomial theorem we see that k is odd for k = 0, 2, 8, 10, and it is even for all other k. Similarly, the product (1 + x)11 ≡ (1 + x8 )(1 + x2 )(1 + x1 )(mod 2) is a polynomial containing 8 = 23 terms, being the product of 3 factors with 2 choices in each. In general, as the sum of p distinct powers ...
... binomial theorem we see that k is odd for k = 0, 2, 8, 10, and it is even for all other k. Similarly, the product (1 + x)11 ≡ (1 + x8 )(1 + x2 )(1 + x1 )(mod 2) is a polynomial containing 8 = 23 terms, being the product of 3 factors with 2 choices in each. In general, as the sum of p distinct powers ...
MACM 101 — Discrete Mathematics I Exercises on Predicates and
... Please, use a pen. 30 points will be taken off for pencil written work. 1. Determine the truth value of each of these statements if the universe of each variable consists of all real numbers. (a) ∃x∃y ((x + 2y = 2) ∧ (2x + 4y = 5)); (b) ∀x∀y∃z (x2 + y 2 = z 2 ). 2. Use predicates and quantifiers to ...
... Please, use a pen. 30 points will be taken off for pencil written work. 1. Determine the truth value of each of these statements if the universe of each variable consists of all real numbers. (a) ∃x∃y ((x + 2y = 2) ∧ (2x + 4y = 5)); (b) ∀x∀y∃z (x2 + y 2 = z 2 ). 2. Use predicates and quantifiers to ...
An Upper Bound on the nth Prime - Mathematical Association of
... In 1845, J. Bertrand conjectured that for any integer n > 3, there exists at least one prime p between n and 2n − 2 [1]. In 1852, P. Tchebychev offered the first demonstration of this now-famous theorem. Today, Bertrand’s Postulate is often stated as, “for any positive integer n ≥ 1, there exists a ...
... In 1845, J. Bertrand conjectured that for any integer n > 3, there exists at least one prime p between n and 2n − 2 [1]. In 1852, P. Tchebychev offered the first demonstration of this now-famous theorem. Today, Bertrand’s Postulate is often stated as, “for any positive integer n ≥ 1, there exists a ...
MATH 327 - Winona State University
... Number of Credits: 4 Catalog Description: With an emphasis on mathematical proof writing, the following topics are covered: structure of the real and complex numbers, elementary number theory, introductory group and field properties, basic topology of the reals. Prerequisite: MATH 242 - Linear Algeb ...
... Number of Credits: 4 Catalog Description: With an emphasis on mathematical proof writing, the following topics are covered: structure of the real and complex numbers, elementary number theory, introductory group and field properties, basic topology of the reals. Prerequisite: MATH 242 - Linear Algeb ...
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.