2 - Set Theory
... S ∪ T = {x | x ∈ S or x ∈ T }. Thus, S ∪ T is the set that contains all the elements in S as well as all the elements in T . We remember, of course, that, the “or” in this definition means that if x ∈ S and also x ∈ T , then it is also included as an element of the union S ∪ T. Notice that, since ev ...
... S ∪ T = {x | x ∈ S or x ∈ T }. Thus, S ∪ T is the set that contains all the elements in S as well as all the elements in T . We remember, of course, that, the “or” in this definition means that if x ∈ S and also x ∈ T , then it is also included as an element of the union S ∪ T. Notice that, since ev ...
Real Analysis - user web page
... of all order pairs (a, b) where a ∈ A and b ∈ B.That is, A X B = { (a ,b) : a ∈ A and b ∈ B }. Example . The Cartesian product of the set of real numbers with it self gives the set of all order pairs of real numbers. We call this set the plane. Definition. Let A and B be any two sets. A function f f ...
... of all order pairs (a, b) where a ∈ A and b ∈ B.That is, A X B = { (a ,b) : a ∈ A and b ∈ B }. Example . The Cartesian product of the set of real numbers with it self gives the set of all order pairs of real numbers. We call this set the plane. Definition. Let A and B be any two sets. A function f f ...
On the Construction of Analytic Sequent Calculi for Sub
... is generalized in the current paper: (1) here we also consider derivations from assumptions (also known as “non-logical axioms”); and (2) we use a more general parametrized notion of a subformula. A particular well-behaved subfamily of pure calculi, called canonical calculi was studied in [4]. For t ...
... is generalized in the current paper: (1) here we also consider derivations from assumptions (also known as “non-logical axioms”); and (2) we use a more general parametrized notion of a subformula. A particular well-behaved subfamily of pure calculi, called canonical calculi was studied in [4]. For t ...
The Cantor Set and the Cantor Function
... Yes, in some sense, a whole lot more. But in some other sense, just some dust - which in some ways is scattered, in some other ways it is bound together. We will describe different ways to ”measure” the dust left. This will take us through several mathematical disciplines: set theory, measure theory ...
... Yes, in some sense, a whole lot more. But in some other sense, just some dust - which in some ways is scattered, in some other ways it is bound together. We will describe different ways to ”measure” the dust left. This will take us through several mathematical disciplines: set theory, measure theory ...
37(2)
... In the previous section, we showed that F(x) generates only even balancing numbers, whereas K(x) generates only odd balancing numbers. But H(x) and K(x) generate both even and odd balancing numbers. Since H(6) = 204 and there is a balancing number 35 between 6 and 204, it is clear that H(x) does not ...
... In the previous section, we showed that F(x) generates only even balancing numbers, whereas K(x) generates only odd balancing numbers. But H(x) and K(x) generate both even and odd balancing numbers. Since H(6) = 204 and there is a balancing number 35 between 6 and 204, it is clear that H(x) does not ...
Catalan Numbers, Their Generalization, and Their Uses
... see the historical note at the end of this ar~cle), which are closely related, both algebraically and conceptually, to the binomial coefficients. These Catalan numbers have many combinatorial interpretations, of which we will emphasize three, in addition to their interpretation in terms of 2-good pa ...
... see the historical note at the end of this ar~cle), which are closely related, both algebraically and conceptually, to the binomial coefficients. These Catalan numbers have many combinatorial interpretations, of which we will emphasize three, in addition to their interpretation in terms of 2-good pa ...
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"".