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MATH 1190 Chapter 4.1,4.3: MATH 1190 Chapters 4.1, 4.3

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greywoodchuck145

29 Jul 2015

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York University

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Mathematics and Statistics

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Professor

Valery Michkine

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The part of mathematics devoted to the study of the set of integers and their properties is known as number theory. Modular arithmetic operates with the remainders of integers when they are divided by a fixed positive integer, called the modulus. Division of an integer by a positive integer produces a quotient and a remainder. Working with these remainders leads to modular arithmetic, which plays an important role in mathematics and which is used throughout computer science. If a and b are integers with a 0, we say that a divides b if there is an integer c such that b = ac, or equivalently, if b/a is an integer. When a divides b we say that a is a factor or divisor of b, and that b is a multiple of a. The notation a | b denotes that a divides b. We write a | b when a does not divide b.

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ey table is seen below xeecions tor Chapter 1 12, Fori, b â¬ Î.el * b-aa (0 (D 14 ForVEQ r the operation defined in Example 1 on the wr John Sue Henry. P iDFor 15ine if is ass ociative by determining if the final six engu defined ang property (ii)of Theorem 1 . So no element i if the operntion debined in Example 1 of Emle if each element has an inverse Section 1.3 Groups iw comutatityn ild Chapter 3 where the Funda the exercises that follow etermining why they are not Write out the Cayley table for the gru(o)nud lensify thee element the 10 elements of the group Z 18 its operation uses ts in the first coordinnte and +2 in the secotu Ldessity the Zn and write out the Cayley table. Reeall th of each element does Theorem 1.14 tell us aont the entries in the Cayley table ol s xrewp that is set with exactly one element. A {n), will always form group since there 19. What nly one way to define an operation on the set. Be sure to shhow that the properi combination of the usual notice what set the rule is be of a group all hold. 21. We can create even create groupe with games! Consider four cups placed in a ssare patter table. If we have a penny in one of the cups there are four ways we can move it to t cup: Horizontally, Vertically, Diagonally, or Stay nehere it is. We will p, S. To define nn operation, consider two movements in a row, v.e.,m ove the penny as r tells us to, then after that move the penny as y instruets. the set of integers the set of real numbers n the set of positive integers. e set of positive integers. n the set of positive integers. on the set of integers e set of negative integers. we ple, HV D since if we first move it horizontally and then vertically nltogether e moved it diagonally. Create the Cayley table for this group, identify the identity and the inverse of cach element 22. Prove that the set A 0 under matrix multiplication. Is it abelian? 23. Verify that 3(R) is a group using the addition of funetions as defined in Example 13 e set of nonzero rational n he set of nonzero real num n on the set R (-1 Section 1.4 Basic Algebra in Groups 24. Prove the Cancellation Law (Theorem 1.20). Do not use any theorems that occur after it in the text ssociative, commutative, has roof or counterexample fo 25. Assume G is a group and a e G. Consider a fixed integer k and use PMI (Theorem 0.3) to prove that for all nEN, a aYou will need to consider cases here since k can be positive, negative, or o 26. Prove Theorem 1.23 (ii) for the case of n > 0 and m

s.pearsoncmg.com/ebook/launcheText.do?values-bookID:47353: platform::1030:bookPage Go CA 114% In this application, we set up a mathematical model for deter.which ean be factored to give mining the total cost required to set up a training program. Thee we use calculus to find the time interval between training pro- grams that produces the minimum total cost. The model assumes The expression in brackets is the sum of the terms of an arithmetic that the demand for trainees is constant and that the fixed cost ofsequence discussed in most algebra texis Using formulas for arid training a batch of trainees is known. Also, it is assumed that metic sequences, we can show that the expression in brackets is people who are trained, but for whom so job is readily availahle. equal to mom-D/2, so we huve will be psid a fixed amount per month while waiting for a job to mim) The model uses the following variables: D demand for trainees per month; N= nunterofumoes per batch C fixed cost of training a batch of trainees Cvariable cost of training per trainee per month ç³.. salay pand monthly a Iraiere who has not yet as the total cos for keeping jobless trainees The total cost per betch is the sum of the training cost per hatch CmDiC, and the cost of keeping trainces without a proper job, given by equsion (X Since we anvsusme hail: batch of trainees is trained every 'n months, the total cost per mont. Zast is given by: been given a job afler training ane interval in months between successive baiches of irainees; ength of training program in months: Zm)ãtotal monthly ast of program The tal cost of training a hasch of trainees is given by Exercises c, .+ MG However, N = al, so ihal lhe total cost per batch is 1. Find Z'o a After training, personnel are given jobs al the tate of D per month ThsNDof the trainees will no gel a job the tirst month. N-2D will no get a joh the second month, and so on The N D rainees who k not get a job the tirst oets proJuce lotial costs of N- DiC hose nos gesting jobs during the second month produce cots of iN-2Co on. Since N-mD.the costs clering the irt month can be writien as 2. Solve the equation Z 0. As a practical matiet, it iND ally required that m be a positive inleger. I m does not come out to be aa integer, then ac and mÂ·the two integers claes to m, masit be choser Calculate both Zim) and onhe sualler the two provides the optimum value er 7 3. Sappose a conpuny finds that its deamand for uainees is three per llknth, that a training program reqtres l2 months. tha.he lised cost of trai ng a balch of inanees is $155xm" ta"he mrginal cost per trainee pet month is St00, and tha riinces are paid $900 per mth after training, but befte going t0 woek. Use your resuls from Exercise 2 to tiedl shilethe costs during the seondmanti-reus .. 20G, andsoon The total cost for keeping the ininees without a jab is thus ances7 How niany traces ould he in a hach? Logal Notice Privacy Policy Permissions Suppoit t Feedback 27

General Math Terms A product B. digit C. numeral D. compass E. equation F. formula G. negative H. sum I. mean J. statistics K. positive L. percent M. straight-edge N. squareroot O. quotient P. prime numbers Q. difference R. addend S. minuend T. divisor Greater than zero: plus. A number being added to another number. A mathematical sequence whose verb is equal (=). The result of addition. The number doing the dividing in a division problem. An instrument used to construct straight lines. A mathematical expression of a natural law. The result of multiplication. The number from which another number is being subtracted. Any non-negative numeral less than ten. Less than zero: minus. An instrument used to construct circles and arcs of circles. A number that produces another number (it's square) when multiplied by itself. The result of subtraction. A symbol or name of a number. The average: the sum of the elements divided by the number of elements. The result of division. Parts per hundred: hundredths. Natural numbers greater than one that have no other factors except one and themselves. Set of numerical data. Geometry A. chord B. diameter C. radius D. perimeter E. cylinder F. surface area G. right angle H. square I. Hypotenuse J. obtuse angle K. vertex L point M. ray N. bisect O arc P. hexagon Q. pentagon R. polygon S. perpendicular T. rectangle U. rhombus V. triangle W. circumference X cube Y pyramid Z. sphere A six-sided polygon. The common end point of the sides of an angle. An angle measuring more than 90 degree. A long, round object with flat, circular ends. An angle of 90 degree. The side of a right triangle opposite the right angle. The total area of all the surfaces of an object. To multiply a number by itself. A location, the place where two lines cross or intersect. A closed geometric figure made up of three or more line segments. The perimeter of a circle: the distance around a circle. A shape having the same dimensions of length, height, and width. A half-line with a designated end point which extends in only one direction without stopping. A figure having a polygon base and triangular sides which meet in a point at the top. Divide an angle into two equal angles. A part of a circle