Section 9.3: Mathematical Induction
... is true. Thus we need to show that a2 = a + (2 − 1)d. Since P (1) is true, we have a1 = a, and by the definition of an arithmetic sequence, a2 = a1 +d = a+d = a+(2−1)d. So P (2) is true. We now use the fact that P (2) is true to show that P (3) is true. Using the fact that a2 = a + (2 − 1)d, we show ...
... is true. Thus we need to show that a2 = a + (2 − 1)d. Since P (1) is true, we have a1 = a, and by the definition of an arithmetic sequence, a2 = a1 +d = a+d = a+(2−1)d. So P (2) is true. We now use the fact that P (2) is true to show that P (3) is true. Using the fact that a2 = a + (2 − 1)d, we show ...
Introduction to Proof in Analysis - 2016 Edition
... The golden rule when writing: never write anything whose meaning is unclear to yourself ! You can also use this text to find many detailed examples of how to write a proof correctly. The language of mathematics consists of assertions about mathematical objects. Mathematical objects include the natur ...
... The golden rule when writing: never write anything whose meaning is unclear to yourself ! You can also use this text to find many detailed examples of how to write a proof correctly. The language of mathematics consists of assertions about mathematical objects. Mathematical objects include the natur ...
Partition of a Set which Contains an Infinite Arithmetic (Respectively
... (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression. Introduction. First, in this article we build sets which have the following property: for any par ...
... (respectively geometric) progression into two subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is an arithmetic (respectively geometric) progression. Introduction. First, in this article we build sets which have the following property: for any par ...
countability diagonalization
... Construct a new bitstring that differs from all others. This new sitstring can’t be in the table. Proof by contradiction: Therefore power(S) must be uncountable! CS340-Discrete Structures ...
... Construct a new bitstring that differs from all others. This new sitstring can’t be in the table. Proof by contradiction: Therefore power(S) must be uncountable! CS340-Discrete Structures ...
Common Core Math Standards
... Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart and comparing with unknowns in all positions. 1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20. 1 ...
... Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart and comparing with unknowns in all positions. 1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20. 1 ...
Guarded negation
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
... as a syntactic fragment of first-order logic, it is also natural to ask for syntactic explanations: what syntactic features of modal formulas (viewed as first-order formulas) are responsible for their good behavior? And can we generalize modal logic, preserving these features, while at the same tim ...
Test - Mu Alpha Theta
... set which is the collection of sets that do not contain its own members. If every set is a subset of itself, than this proves contradictory. What is the name of this idea? (a) Cantor's paradox (d) Empty Set contradiction (b) Russell's paradox (e) None of the Above (c) Unit Set theory 18. If Set A = ...
... set which is the collection of sets that do not contain its own members. If every set is a subset of itself, than this proves contradictory. What is the name of this idea? (a) Cantor's paradox (d) Empty Set contradiction (b) Russell's paradox (e) None of the Above (c) Unit Set theory 18. If Set A = ...
Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson. He wrote:[...] the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection [...] that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely large compared with the real numbers but which were to possess the same properties as the latterRobinson argued that this law of continuity of Leibniz's is a precursor of the transfer principle. Robinson continued:However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort. As a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits.Robinson continues:It is shown in this book that Leibniz's ideas can be fully vindicated and that they lead to a novel and fruitful approach to classical Analysis and to many other branches of mathematics. The key to our method is provided by the detailed analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory.In 1973, intuitionist Arend Heyting praised non-standard analysis as ""a standard model of important mathematical research"".