Cardinality, countable and uncountable sets
... two finite sets have the same number of elements: we just need to verify their elements can be placed in pairwise correspondence; that is, that there is a bijection between them. It is then natural to generalize this to infinite sets, and indeed to arbitrary sets. Definition 1. Let A, B be sets. We ...
... two finite sets have the same number of elements: we just need to verify their elements can be placed in pairwise correspondence; that is, that there is a bijection between them. It is then natural to generalize this to infinite sets, and indeed to arbitrary sets. Definition 1. Let A, B be sets. We ...
Discussion
... below. The unit squares also form a triangular shape that can be copied and rotated 180 degrees to form a rectangle. One-half the area of this n by n + 1 rectangle is the sum of the first n natural numbers. ...
... below. The unit squares also form a triangular shape that can be copied and rotated 180 degrees to form a rectangle. One-half the area of this n by n + 1 rectangle is the sum of the first n natural numbers. ...
Numbers Which Factor as Their Digital Sum Times a Prime
... This problem is a particular case of an open problem discussed at length in the recent book of De Koninck [3, page 75], which is certainly very hard to prove despite the fact that it is supported by numerical evidence. Instead of proving Problem 1, it is very likely that one may discover other const ...
... This problem is a particular case of an open problem discussed at length in the recent book of De Koninck [3, page 75], which is certainly very hard to prove despite the fact that it is supported by numerical evidence. Instead of proving Problem 1, it is very likely that one may discover other const ...
Lecture 12
... • Take the case of ten people in a row: there are 10 choices for the first person; then, since we’ve chosen the first person, there are 9 choices for the second; then 8 choices for the third; and so forth. So overall, there are 10! (= 10 * 9 * 8 * …. 1) ways of ...
... • Take the case of ten people in a row: there are 10 choices for the first person; then, since we’ve chosen the first person, there are 9 choices for the second; then 8 choices for the third; and so forth. So overall, there are 10! (= 10 * 9 * 8 * …. 1) ways of ...
KEY CONCEPT
... NUMBER LINE Real numbers can be graphed as points on a line called a real number line, on which numbers increase from left to right. ...
... NUMBER LINE Real numbers can be graphed as points on a line called a real number line, on which numbers increase from left to right. ...
Excel Learning Sheet
... Attempt the following activities to develop your skills using a spreadsheet. You may find much of this is very applicable to your project. For assistance, you can learn about EXCEL functions by clicking the ’fx’ button/icon just over column C. Simulate Tossing a Coin Use Column A and B to show a ser ...
... Attempt the following activities to develop your skills using a spreadsheet. You may find much of this is very applicable to your project. For assistance, you can learn about EXCEL functions by clicking the ’fx’ button/icon just over column C. Simulate Tossing a Coin Use Column A and B to show a ser ...
Lecture notes 2 -- Sets
... communicate them, and what we can do with them. As we will see over the next few months, sets show up everywhere in mathematics. The good news is that as far as we’re concerned, they are fairly easy to describe. The term set in mathematics pretty much means what you probably think it means: a collec ...
... communicate them, and what we can do with them. As we will see over the next few months, sets show up everywhere in mathematics. The good news is that as far as we’re concerned, they are fairly easy to describe. The term set in mathematics pretty much means what you probably think it means: a collec ...
Math311W08Day3
... Similarly, the sequence cannot converge to 0 since (with =1) for any natural numberthere is a larger even number (so there is some n N where the value of the sequence is 2). Thus this term is not within epsilon of zero. So the sequence has no ...
... Similarly, the sequence cannot converge to 0 since (with =1) for any natural numberthere is a larger even number (so there is some n N where the value of the sequence is 2). Thus this term is not within epsilon of zero. So the sequence has no ...
Binet`s formula for Fibonacci numbers
... Our interest in Fibonacci numbers stems from their frequent occurrence in nature. Take flowers, for example. Consistently, the number of petals in a daisy is a Fibonacci number, which depends on the variety: 13 for Blue daisies; 21 for English daisies; 34 for Oxeye daisies; 55 for African daisies, a ...
... Our interest in Fibonacci numbers stems from their frequent occurrence in nature. Take flowers, for example. Consistently, the number of petals in a daisy is a Fibonacci number, which depends on the variety: 13 for Blue daisies; 21 for English daisies; 34 for Oxeye daisies; 55 for African daisies, a ...
Formal power series
... And we can proceed from there to find an exact formula. But rather than do it that way, let’s start the problem over. My method: Even before we know what the initial conditions are, we can check that the recurrence relation gives (1-4x+3x^2) F(x) = (1-4x+3x^2)(f(0)+f(1)x+f(2)x^2+f(3)x^3+...) = [f(0) ...
... And we can proceed from there to find an exact formula. But rather than do it that way, let’s start the problem over. My method: Even before we know what the initial conditions are, we can check that the recurrence relation gives (1-4x+3x^2) F(x) = (1-4x+3x^2)(f(0)+f(1)x+f(2)x^2+f(3)x^3+...) = [f(0) ...
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"".