THE CHINESE REMAINDER THEOREM

The Chinese remainder theorem can be extended from two congruences to an arbitrary nite number of congruences, but we have to be careful about the way in which the moduli are relatively prime. Consider the three congruences x 1 mod 6; x 4 mod 10; x 7 mod 15:

  Chinese, Theorem, Remainder, Chinese remainder theorem

DIFFERENTIATING UNDER THE INTEGRAL SIGN - Keith …

2 KEITH CONRAD If you are used to thinking mostly about functions with one variable, not two, keep in mind that (1.2) involves integrals and derivatives with respect to separate variables: integration with respect to xand di erentiation with respect to t. Example 1.2. We saw in Example1.1that R 1 0 (2x+t3)2 dx= 4=3+2t3 +t6, whose t-derivative ...

  Conrad, Relating

Introduction The Miller{Rabin test - Keith Conrad's Home Page

KEITH CONRAD 1. Introduction The Miller{Rabin test is the most widely used probabilistic primality test. For odd composite n>1 over 75% of numbers from to 2 to n 1 are witnesses in the Miller{Rabin test for n. We will describe the test, prove the …

  Conrad, Miller, Brain, Miller rabin

GALOIS THEORY AT WORK: CONCRETE EXAMPLES

GALOIS THEORY AT WORK: CONCRETE EXAMPLES 3 Remark 1.3. While Galois theory provides the most systematic method to nd intermedi-ate elds, it may be possible to argue in other ways. For example, suppose Q ˆFˆQ(4 p 2) with [F: Q] = 2. Then 4 p 2 has degree 2 over F. Since 4 p 2 is a root of X4 2, its minimal polynomial over Fhas to be a ...

  Work, Galois

Construction - University of Connecticut

k, which are all multiples of pand thus vanish in F). Therefore the pth power map t7!tpon Fis additive. The map t7!tpn is also additive since it’s the n-fold composite of t7!tp with itself and the composition. 4 KEITH CONRAD of homomorphisms is a homomorphism.2 The xed points of an additive map are a group

ORDERS OF ELEMENTS IN A GROUP Introduction

Theorem3.2gives a nice combinatorial interpretation of the order of g, when it is nite: the order of gis the size of the group hgi. In fact, this even works when ghas in nite order (then hgiis an in nite group), so the order of gis always the size of hgi. The nite order of an element is linked to periodicity in its powers, as follows. Theorem 3.4.

  Introduction, Order, Combinatorial

TENSOR PRODUCTS Introduction R e f i;j c e f

Tensor products rst arose for vector spaces, and this is the only setting where they occur in physics and engineering, so we’ll describe tensor products of vector spaces rst. Let V and W be vector spaces over a eld K, and choose bases fe igfor V and ff jgfor W. The tensor product V KWis de ned to be the K-vector space with a basis of formal ...

  Product, Vector, Tensor, Tensor product

DIHEDRAL GROUPS Introduction - University of Connecticut

1. Introduction For n 3, the dihedral group D n is de ned as the rigid motions1 taking a regular n-gon back to itself, with the operation being composition. These polygons for n= 3;4, 5, and 6 are pictured below. The dotted lines are lines of re ection: re ecting the polygon across each line brings the polygon back to itself, so these re ...

  Introduction, Group, Dihedral, Dihedral group

GRAPHS OF FUNCTIONS AND DERIVATIVES

GRAPHS OF FUNCTIONS AND DERIVATIVES 3 x y a b Figure 5. A continuous function on (a;b). x y a Figure 6. A continuous function on [a;1). x y …

  Derivatives

Direct Proofs - Stanford University

Direct Proofs A direct proof is the simplest type of proof. Starting with an initial set of assumptions, apply simple logical steps to derive the result. Directly prove that the result is true. Contrasts with indirect proofs, which we'll see on Friday.

  Proof

Introduction to mathematical arguments

prove any type of statement. This chart does not include uniqueness proofs and proof by induction, which are explained in §3.3 and §4. Apendix A reviews some terminology from set theory which we will use and gives some more (not terribly interesting) examples of proofs. 1

  Proof

Logic, Sets, and Proofs - Amherst College

A set is a collection of objects, which are called elements or members of the set. Two sets are equal when they have the same elements. ... strategies for di erent types of proofs. Direct Proof. The simplest way to prove A )B is to assume A (the \hypothe-sis") and prove B (the \conclusion"). See Proof 2 in Section 5 for a direct proof of

  College, Proof, Amherst, Amherst college

A GUIDE TO PROOFS IN LINEAR ALGEBRA

Logical deduction was the fourth element in our list of ingredients for writing proofs. Much of our logical structure is buried in the development of axiomatic structure and set theory. From this we get the theorems we’ve previously developed in mathematics such as Euclidean geometry, algebra, trigonometry, and calculus.

  Linear, Proof, Algebra, Linear algebra

Logic, Sets, and Proofs - Amherst

Logic, Sets, and Proofs David A. Cox and Catherine C. McGeoch Amherst College 1 Logic Logical Operators. A logical statement is a mathematical statement that can be ... A set is a collection of objects, which are called elements or members of the set. Two sets are equal when they have the same elements.

  Proof, Amherst

Basic Proofs - Loyola University Maryland

There are four basic proof techniques to prove p =)q, where p is the hypothesis (or set of hypotheses) and q is the result. 1.Direct proof 2.Contrapositive 3.Contradiction 4.Mathematical Induction What follows are some simple examples of proofs. You very likely saw these in MA395: Discrete Methods. 1 Direct Proof

  Proof

Proof Techniques - Stanford University Computer Science

monly seen in proofs. 1.1.1 Proof by contrapositive Consider the statement \If it is raining today, then I do not go to class." This is logically equivalent to the statement \If I go to class, then it is not raining today." So if we want to prove the rst statement, it su ces to prove the second statement (which is called the contrapositive).

  Proof

Chapter 1. Metric spaces - Proofs covered in class

5 A set is open if and only if it is a union of open balls. Proof. Suppose first that U is a union of open balls. Then U is a union of open sets by part 4, so it is open itself by part 2. Conversely, suppose that U is an open set. Given any x ∈ U, we can then find some ε x > 0such that B(x,ε x)⊂ U. This gives {x} ⊂ B(x,ε x)⊂ U

  Proof

Identity 2. - gatech.edu

Further, x ∈ A and x 6∈B also by definition of set difference. Thus x ∈ A and x 6∈B and x 6∈C, which implies x 6∈(BorC). Hence, x 6∈(B∪C) by definition of union. Thus, given x ∈ A we have x ∈ A−(B∪C) by definition of set difference. 1. Identity 4. Let A, B and C be sets. Show that