MMExternalRepresentations
... A function may be given by a formula. You may display the graph of a continuous function on the reals. Example. A function may be given by a table of values. A linear transformation on a finite dimensional vector space can be represented by a matrix. The chapter on images and metaphors for f ...
... A function may be given by a formula. You may display the graph of a continuous function on the reals. Example. A function may be given by a table of values. A linear transformation on a finite dimensional vector space can be represented by a matrix. The chapter on images and metaphors for f ...
Outline of Lecture 2 First Order Logic and Second Order Logic Basic
... a biggest (smallest) element smaller (larger) than a) • a dense linear order (between any two distinct elements there is a further element); Using compactness we can prove that following are not FOL-definable: • A is finite, A is countable; • a well-ordering (every subset of A has a least element); ...
... a biggest (smallest) element smaller (larger) than a) • a dense linear order (between any two distinct elements there is a further element); Using compactness we can prove that following are not FOL-definable: • A is finite, A is countable; • a well-ordering (every subset of A has a least element); ...
Common Core Standards
... Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in t ...
... Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in t ...
4.6: Formalizing Relations and Functions
... ordered pairs (x, y). In this case, the domain is the set of x-values and the range is the set of y-values. ...
... ordered pairs (x, y). In this case, the domain is the set of x-values and the range is the set of y-values. ...
6 Post Notes
... f) The sine and cosine functions are continuous over all real numbers. g) The cotangent, cosecant, secant and tangent functions are continuous over their domain. More continuous functions If f(x) and g(x) are continuous at x = c, then so are ...
... f) The sine and cosine functions are continuous over all real numbers. g) The cotangent, cosecant, secant and tangent functions are continuous over their domain. More continuous functions If f(x) and g(x) are continuous at x = c, then so are ...
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. In 1927, it appeared in a second edition with an important Introduction To the Second Edition, an Appendix A that replaced ✸9 and an all-new Appendix C.PM, as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven. As such, this ambitious project is of great importance in the history of mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem proved definitively that PM, and in fact any other attempt, could never achieve this lofty goal; that is, for any set of axioms and inference rules proposed to encapsulate mathematics, either the system must be inconsistent, or there must in fact be some truths of mathematics which could not be deduced from them.One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. This was achieved by replacing the notion of a general set with the notion of a hierarchy of sets of different 'types', a set of a certain type only allowed to contain sets of strictly lower types. Contemporary mathematics, however, avoids paradoxes such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory.PM is not to be confused with Russell's 1903 Principles of Mathematics. PM states: ""The present work was originally intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became increasingly evident that the subject is a very much larger one than we had supposed; moreover on many fundamental questions which had been left obscure and doubtful in the former work, we have now arrived at what we believe to be satisfactory solutions.""The Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century.