Mathematical Proofs - Kutztown University
... Example: The set of all positive even integers less than 41 can be described by X={2, 4, …, 40} The set of all positive odd integers can be described by Y={1, 3, 5, …} ...
... Example: The set of all positive even integers less than 41 can be described by X={2, 4, …, 40} The set of all positive odd integers can be described by Y={1, 3, 5, …} ...
ASSIGNMENT 3
... example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined. 2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is accepted without proof. For example, the statement “A straight line can be drawn from any point to any point” ...
... example, in modern Euclidean geometry the terms ‘point and ‘line’ are typically left undefined. 2. Axioms (or Postulates): An axiom (or postulate) is a logical statement about terms that is accepted without proof. For example, the statement “A straight line can be drawn from any point to any point” ...
{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A
... you are talking about) is called the “UNIVERSE” and is represented by the symbol: ...
... you are talking about) is called the “UNIVERSE” and is represented by the symbol: ...
timeline
... Russell's first major book in its philosophical style: Our knowledge of the external world as a field for scientific method in philosophy March-May 1914 lecture courses by Russell at Harvard University and the Lowell Institute on Principia mathematica and on Our knowledge; extensive notes ...
... Russell's first major book in its philosophical style: Our knowledge of the external world as a field for scientific method in philosophy March-May 1914 lecture courses by Russell at Harvard University and the Lowell Institute on Principia mathematica and on Our knowledge; extensive notes ...
Induction
... up to 63 pounds if one is allowed to put weights in only one pan of a balance? 10) Prove that every natural number has a unique representation in each base. ...
... up to 63 pounds if one is allowed to put weights in only one pan of a balance? 10) Prove that every natural number has a unique representation in each base. ...
Combining Signed Numbers
... The easiest way to describe many sets is by using words. B = {all blue-eyed blonds in this class} There isn’t a really good mathematical equivalent for that! ...
... The easiest way to describe many sets is by using words. B = {all blue-eyed blonds in this class} There isn’t a really good mathematical equivalent for that! ...
Mathematical Ideas - Millersville University of Pennsylvania
... © 2008 Pearson Addison-Wesley. All rights reserved ...
... © 2008 Pearson Addison-Wesley. All rights reserved ...
The Closed World Assumption
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
... We view our program as a logical theory expressing knowledge about the world. In several situations, it is convenient to assume that the program contains complete information about certain kinds of logical statements. We can then make additional inferences about the world based on the assumed comple ...
Set-Builder Notation
... (called curly braces) – We use … (ellipses) to denote a set extending infinitely in the same pattern • The set of even numbers can then be expressed as {0, ...
... (called curly braces) – We use … (ellipses) to denote a set extending infinitely in the same pattern • The set of even numbers can then be expressed as {0, ...
mplications of Cantorian Transfinite Set Theory
... David Hilbert described Cantor's work as:“...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” "I see it but I don't believe it.” Georg Cantor on his own theory. “…the infinite is nowhere to be found in reality” David Hilbert. ...
... David Hilbert described Cantor's work as:“...the finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.” "I see it but I don't believe it.” Georg Cantor on his own theory. “…the infinite is nowhere to be found in reality” David Hilbert. ...
CS311H: Discrete Mathematics Cardinality of Infinite Sets and
... First list those with p + q = 2, then p + q = 3, . . . ...
... First list those with p + q = 2, then p + q = 3, . . . ...
MATH 2420 Discrete Mathematics
... to as the power set of a set and is denoted P(§). But how many sets are there in the power set? The number of sets is equal to 2n where n is the number of elements in a set. For example, if we have a set A = {2, 4, 6} then the power set P(A) consists of ...
... to as the power set of a set and is denoted P(§). But how many sets are there in the power set? The number of sets is equal to 2n where n is the number of elements in a set. For example, if we have a set A = {2, 4, 6} then the power set P(A) consists of ...
Set theory
Set theory is the branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects.The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with the axiom of choice, are the best-known.Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.