Real Analysis - University of Illinois at Chicago
... In Chapter 0 we review the basic ideas of mathematics and logical reasoning needed to complete the study. Like Euclid’s Basic Notions, these are the things about sets and logic that we hold to be self-evident and natural for gluing together formal arguments of proof. This chapter can be covered sepa ...
... In Chapter 0 we review the basic ideas of mathematics and logical reasoning needed to complete the study. Like Euclid’s Basic Notions, these are the things about sets and logic that we hold to be self-evident and natural for gluing together formal arguments of proof. This chapter can be covered sepa ...
6•3 Lesson 1 Problem Set
... The fuel gauge in Nic’s car says that he has 26 miles to go until his tank is empty. He passed a fuel station 19 miles ago and a sign says there is a town only 8 miles ahead. If he takes a chance and drives ahead to the town and there isn’t a fuel station there, does he have enough fuel to go back t ...
... The fuel gauge in Nic’s car says that he has 26 miles to go until his tank is empty. He passed a fuel station 19 miles ago and a sign says there is a town only 8 miles ahead. If he takes a chance and drives ahead to the town and there isn’t a fuel station there, does he have enough fuel to go back t ...
Types of Numbers - SD43 Teacher Sites
... The sum of the cubes of the first n odd numbers is 2n4 - n2 = n2(2n2 - 1). The sum of the cubes of the first n even numbers is 2n4 + 4n3 + 2n2 = 2n2(n + 1)2. The sum of the first n cubes, 13 + 23 + 33 + 43 +........+ n3 is equal to the square of the sum of the first n integers. Thus, 13 + 23 + 33 + ...
... The sum of the cubes of the first n odd numbers is 2n4 - n2 = n2(2n2 - 1). The sum of the cubes of the first n even numbers is 2n4 + 4n3 + 2n2 = 2n2(n + 1)2. The sum of the first n cubes, 13 + 23 + 33 + 43 +........+ n3 is equal to the square of the sum of the first n integers. Thus, 13 + 23 + 33 + ...
The Connectedness of Arithmetic Progressions in
... arithmetic progressions in Golomb’s topology D on N. Theorem 3.3. Let a, b ∈ N. The arithmetic progression {an + b} is connected in (N, D) if and only if Θ(a) ⊆ Θ(b). In particular, i) the progression {an} is D-connected, and ii) if the progression {an + b} is an element of the basis B, then it is D ...
... arithmetic progressions in Golomb’s topology D on N. Theorem 3.3. Let a, b ∈ N. The arithmetic progression {an + b} is connected in (N, D) if and only if Θ(a) ⊆ Θ(b). In particular, i) the progression {an} is D-connected, and ii) if the progression {an + b} is an element of the basis B, then it is D ...
On the representation of integers as sums of triangular number
... using modular form theory. Incidentally, Gauss’ famous Eureka theorem asserts that every non-negative integer is represented as a sum of three triangular numbers. In our notation this says that if n ≥ 0, then δ3 (n) > 0. The reader may consult [1] for a discussion of this theorem. Now we give some b ...
... using modular form theory. Incidentally, Gauss’ famous Eureka theorem asserts that every non-negative integer is represented as a sum of three triangular numbers. In our notation this says that if n ≥ 0, then δ3 (n) > 0. The reader may consult [1] for a discussion of this theorem. Now we give some b ...
Predicate Logic for Software Engineering
... Since under conventional interpretations a logical expression that includes partial functions will have a defined value only when the values assigned to all function arguments are within functions’ domains. Such interpretations are of limited usefulness when describing software because we frequent ...
... Since under conventional interpretations a logical expression that includes partial functions will have a defined value only when the values assigned to all function arguments are within functions’ domains. Such interpretations are of limited usefulness when describing software because we frequent ...
A Basis for a Mathematical Theory of Computation
... subscripts) for variables and will suppose that there is a notation for constants that does not make expressions ambiguous. (Thus, the decimal notation is allowed for constants when we are dealing with integers.) The class of forms is defined recursively as follows: (i) A variable x with an associat ...
... subscripts) for variables and will suppose that there is a notation for constants that does not make expressions ambiguous. (Thus, the decimal notation is allowed for constants when we are dealing with integers.) The class of forms is defined recursively as follows: (i) A variable x with an associat ...
