DISCRETE MATHEMATICAL STRUCTURES
... Conditional Propositions: A proposition of the form ―if p then q‖ or ―p implies q‖, represented ―p → q‖ is called a conditional proposition. For instance: ―if John is from Chicago then John is from Illinois‖. The proposition p is called hypothesis or antecedent, and the proposition q is the conclusi ...
... Conditional Propositions: A proposition of the form ―if p then q‖ or ―p implies q‖, represented ―p → q‖ is called a conditional proposition. For instance: ―if John is from Chicago then John is from Illinois‖. The proposition p is called hypothesis or antecedent, and the proposition q is the conclusi ...
minimum models: reasoning and automation
... a minimum model, as presented in [1] and [2], appears naturally when modelling systems whose states can be described by positive “facts” or “observations” (many applications in Artificial Intelligence use this kind of representation for states, see [6], [9], [18]) and where observations not stated o ...
... a minimum model, as presented in [1] and [2], appears naturally when modelling systems whose states can be described by positive “facts” or “observations” (many applications in Artificial Intelligence use this kind of representation for states, see [6], [9], [18]) and where observations not stated o ...
What every computer scientist should know about floating
... bit. The precise encoding is not important for now. There are two reasons why a real number might not be exactly representable as a floating-point number. The most common situation is illustrated by the decimal number 0.1. Although it has a finite decimal representation, in binary it has an infinite ...
... bit. The precise encoding is not important for now. There are two reasons why a real number might not be exactly representable as a floating-point number. The most common situation is illustrated by the decimal number 0.1. Although it has a finite decimal representation, in binary it has an infinite ...
Floating-Point Arithmetic Goldberg CS1991
... bit. The precise encoding is not important for now. There are two reasons why a real number might not be exactly representable as a floating-point number. The most common situation is illustrated by the decimal number 0.1. Although it has a finite decimal representation, in binary it has an infinite ...
... bit. The precise encoding is not important for now. There are two reasons why a real number might not be exactly representable as a floating-point number. The most common situation is illustrated by the decimal number 0.1. Although it has a finite decimal representation, in binary it has an infinite ...
39(5)
... drawn in India ink on separate sheets of bond paper or vellum, approximately twice the size they are to appear in print. Since the Fibonacci Association has adopted ¥{ = F2 = 1, F« +;= F« +F«-;, n>2 and L ^ l , L2 =3, L/2+/ = Ln+Ln-i, n>2 as the standard definitions for The Fibonacci and Lucas seque ...
... drawn in India ink on separate sheets of bond paper or vellum, approximately twice the size they are to appear in print. Since the Fibonacci Association has adopted ¥{ = F2 = 1, F« +;= F« +F«-;, n>2 and L ^ l , L2 =3, L/2+/ = Ln+Ln-i, n>2 as the standard definitions for The Fibonacci and Lucas seque ...
Proof, Sets, and Logic - Boise State University
... 4/5/2013: Considering subversive language about second-order logic. Where to put it? I added a couple of sections with musings about second order logic. They are probably not in the right places, but they might be modified to fit where they are or moved to better locations. November 30, 2012: readi ...
... 4/5/2013: Considering subversive language about second-order logic. Where to put it? I added a couple of sections with musings about second order logic. They are probably not in the right places, but they might be modified to fit where they are or moved to better locations. November 30, 2012: readi ...
41(2)
... The first assertion we shall disprove states that there are infinitely many pairs of positive coprime integers x, y such that 2\y, x2 + y2 E D, and ...
... The first assertion we shall disprove states that there are infinitely many pairs of positive coprime integers x, y such that 2\y, x2 + y2 E D, and ...
ARE THERE INFINITELY MANY TWIN PRIMES
... to conclude that this is not just an empirical observation, but a fact built into the structure of whole numbers themselves. It was the genius of the ancient Greeks to develop mathematics not just as an empirical science, but as an axiomatic system for logical deductions.2 In Euclid we find in place ...
... to conclude that this is not just an empirical observation, but a fact built into the structure of whole numbers themselves. It was the genius of the ancient Greeks to develop mathematics not just as an empirical science, but as an axiomatic system for logical deductions.2 In Euclid we find in place ...
An Introduction to Higher Mathematics
... logical expressions similar to the algebra for numerical expressions. This subject is called Boolean Algebra and has many uses, particularly in computer science. If two formulas always take on the same truth value no matter what elements from the universe of discourse we substitute for the various v ...
... logical expressions similar to the algebra for numerical expressions. This subject is called Boolean Algebra and has many uses, particularly in computer science. If two formulas always take on the same truth value no matter what elements from the universe of discourse we substitute for the various v ...
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"".