Chapter Prove or disprove: A (B C) = (A B) (A C). Ans: True, since


 Jacob Nelson
 1 years ago
 Views:
Transcription
1 Chapter 2 1. Prove or disprove: A (B C) = (A B) (A C)., since A ( B C) = A B C = A ( B C) = ( A B) ( A C) = ( A B) ( A C). 2. Prove that A B= A B by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side). Ans: A B A B : Let x A B. x A B, x A or x B, x A or x B, x A B. Reversing the steps shows that A B A B. 3. Prove that A B= A B by giving an element table proof. Ans: The columns for A B and A B match: each entry is 0 if and only if A and B have the value Prove that A B= A B by giving a proof using logical equivalence. A B={ x x A B} Ans: ={ xx A B} ={ x ( x A B) } ={ x ( x A x B) } ={ x ( x A) ( x B) } ={ xx A x B} ={ xx A x B} ={ x x A B}= A B 5. Prove that A B = A B by giving a Venn diagram proof. Ans:. Page 22
2 6. Prove that A (B C) = (A B) (A C) by giving a containment proof (that is, prove that the left side is a subset of the right side and that the right side is a subset of the left side). Ans: A (B C) (A B) (A C): Let x A (B C). x A and x B C, x A and x B, or x A and x C, x (A B) (A C). Reversing the steps gives the opposite containment. 7. Prove that A (B C) = (A B) (A C) by giving an element table proof. Ans: Each set has the same values in the element table: the value is 1 if and only if A has the value 1 and either B or C has the value Prove that A (B C) = (A B) (A C) by giving a proof using logical equivalence. A ( B C) ={ x x A ( B C) } Ans: ={ xx A x ( B C) } ={ xx A ( x B x C) } ={ x ( x A x B) ( x A x C) } ={ xx A B x A C} ={ xx ( A B) ( A C) } = ( A B) ( A C) 9. Prove that A (B C) = (A B) (A C) by giving a Venn diagram proof. Ans: Page 23
3 10. Prove or disprove: if A, B, and C are sets, then A (B C) = (A B) (A C).. For example, let A = {1,2}, B = {1}, C = {2}. 11. Prove or disprove A (B C) = (A B) C., using either a membership table or a containment proof, for example. Use the following to answer questions 1215: Use a Venn diagram to determine which relationship,, =,, is true for the pair of sets. 12. A B, A (B A). Ans: =. 13. A (B C), (A B) C. Ans:. 14. (A B) (A C), A (B C). Ans: =. 15. (A C) (B C), A B. Ans:. Use the following to answer questions 1620: In the questions below determine whether the given set is the power set of some set. If the set is a power set, give the set of which it is a power set. 16. {,{ },{a},{{a}},{{{a}}},{,a},{,{a}},{,{{a}}},{a,{a}},{a,{{a}}},{{a},{{a}}}, {,a,{a}},{,a,{{a}}},{,{a},{{a}}},{a,{a},{{a}}},{,a,{a},{{a}}}}. Ans: Yes {,a,{a},{{a}}}. 17. {,{a}}. Ans: Yes, {a}. 18. {,{a},{,a}}. Ans: No, it lacks { }. 19. {,{a},{ },{a, }}. Ans: Yes, {{a, }}. 20. {,{a, }}. Ans: No, it lacks {a} and { }. Page 24
4 21. Prove that S T = S T for all sets S and T. Ans: Since S T = S T (De Morgan's law), the complements are equal. Use the following to answer questions 2233: In the questions below mark each statement TRUE or FALSE. Assume that the statement applies to all sets. 22. A (B C) = (A B) C. 23. (A C) (B C) = A B. 24. A (B C) = (A B) (A C). 25. A (B C) = (A B) (A C). 26. A B A= A. 27. If A C = B C, then A = B. 28. If A C = B C, then A = B. 29. If A B = A B, then A = B. 30. If A B = A, then B = A. 31. There is a set A such that P(A) = A A = A. 33. Find three subsets of {1,2,3,4,5,6,7,8,9} such that the intersection of any two has size 2 and the intersection of all three has size 1. Ans: For example, {1,2,3}, {2,3,4}, {1,3,4}. Page 25
5 + 34. Find [ 1/ i,1/ i]. i = 1 Ans: [1,1] Find ( 1, 1). i i = 1 Ans: Find [ 1, 1]. i i = 1 Ans: {1} Find ( i, ). i = 1 Ans:. 38. Suppose U = {1, 2,..., 9}, A = all multiples of 2, B = all multiples of 3, and C = {3, 4, 5, 6, 7}. Find C  (B  A). Ans: {4,5,6,7}. 39. Suppose S = {1, 2, 3, 4, 5}. Find PS ( ). Ans: 32. Use the following to answer questions 4043: In the questions below suppose A = {x,y} and B = {x,{x}}. Mark the statement TRUE or FALSE 40. x B. 41. P(B). 42. {x} A B. 43. P(A) = 4. Page 26
6 Use the following to answer questions 4451: In the questions below suppose A = {a,b,c}. Mark the statement TRUE or FALSE. 44. {b,c} P(A). 45. {{a}} P(A). 46. A. 47. { } P(A). 48. A A. 49. {a,c} A. 50. {a,b} A A. 51. (c,c) A A. Use the following to answer questions 5259: In the questions below suppose A = {1,2,3,4,5}. Mark the statement TRUE or FALSE. 52. {1} P(A). 53. {{3}} P(A). 54. A. 55. { } P(A). Page 27
7 56. P(A). 57. {2,4} A A. 58. { } P(A). 59. (1,1) A A. Use the following to answer questions 6062: In the questions below suppose the following are fuzzy sets: F = {0.7Ann,0.1Bill,0.8Fran,0.3Olive,0.5Tom}, R = {0.4Ann,0.9Bill,0.9Fran,0.6Olive,0.7Tom}. 60. Find F and R. Ans: F = { 03. Ann, 09. Bill, 02. Fran, 07. Olive, 05. Tom}, R ={ 06. Ann, 01. Bill, 01. Fran, 04. Olive, 03. Tom} 61. Find F R. Ans: {0.7Ann,0.9Bill,0.9Fran,0.6Olive,0.7Tom}. 62. Find F R. Ans: {0.4Ann,0.1Bill,0.8Fran,0.3Olive,0.5Tom}. Use the following to answer questions 6372: In the questions below, suppose A = {a,b,c} and B = {b,{c}}. Mark the statement TRUE or FALSE 63. c A B. 64. P(A B) = P(B). 66. B A. Page 28
8 67. {c} B. 68. {a,b} A A. 69. {b,c} P(A). 70. {b,{c}} P(B). 71. A A. 72. {{{c}}} P(B). Use the following to answer questions 7385: In the questions below determine whether the set is finite or infinite. If the set is finite, find its size. 73. {x x Z and x 2 < 10}. Ans: P({a,b,c,d}), where P denotes the power set. Ans: {1,3,5,7, }. Ans: Infinite. 76. A B, where A = {1,2,3,4,5} and B = {1,2,3}. Ans: {x x N and 9x 2 1 = 0}. Ans: P(A), where A is the power set of {a,b,c}. Ans: A B, where A = {a,b,c} and B =. Ans: 0. Page 29
9 80. {x x N and 4x 2 8 = 0}. Ans: {x x Z and x 2 = 2}. Ans: P(A), where A = P({1,2}). Ans: {1,10,100,1000, }. Ans: Infinite. 84. S T, where S = {a,b,c} and T = {1,2,3,4,5}. Ans: {x x Z and x 2 < 8}. Ans: Prove that between every two rational numbers a/b and c/d (a) there is a rational number. (b) there are an infinite number of rational numbers. Ans: (a) Assume a a c c a b + d ad + bc c. Then =. b d b 2 2bd d (b) Assume a c a c <. Let m 1 be the midpoint of b d, b d a mi midpoint of, 1.. b 87. Prove that there is no smallest positive rational number. a a a a Ans: If 0 <, then 0 < < < < < a b 4 b 3 b 2 b b. Use the following to answer questions 8896:. For i > 1 let m i be the In the questions below determine whether the rule describes a function with the given domain and codomain. 88. f : N N where f ( n) = n. Ans: Not a function; f(2) is not an integer. 89. h : R R where hx ( ) = x. Ans: Function. Page 30
10 90. g : N N where g(n) = any integer > n. Ans: Not a function; g(1) has more than one value F : R R where F( x) =. x 5 Ans: Not a function; F(5) not defined F : Z R where F( x) = 2 x 5 Ans: Function F : Z Z where F( x) =. 2 x 5 Ans: Not a function; F(1) not an integer. x+ 2 if x G : R R where G( x) = x 1 if x 4 Ans: Not a function; the cases overlap. For example, G(1) is equal to both 3 and 0. 2 x if x f : R R where f( x) = x 1 if x 4 Ans: Not a function; f(3) not defined. 96. G : Q Q where G(p/q) = q. Ans: Not a function; f(1/2) = 2 and f(2/4) = Give an example of a function f : Z Z that is 11 and not onto Z. Ans: f(n) = 2n. 98. Give an example of a function f : Z Z that is onto Z but not 11. n Ans: f( n ) = Give an example of a function f : Z N that is both 11 and onto N. 2 n if n 0 Ans: f( n ) = 2n 1 if x> 0 Page 31
11 100. Give an example of a function f : N Z that is both 11 and onto Z. n, neven 2 Ans: f(n) = 2 n + 1, n odd Give an example of a function f : Z N that is 11 and not onto N. 2 n, n 0 Ans: f( n ) = 2n+ 1, n> Give an example of a function f : N Z that is onto Z and not 11. n, n odd Ans: f( n ) = 2 n1, n even Suppose f : N N has the rule f(n) = 4n + 1. Determine whether f is 11. Ans: Yes Suppose f : N N has the rule f(n) = 4n + 1. Determine whether f is onto N. Ans: No Suppose f : Z Z has the rule f(n) = 3n 2 1. Determine whether f is 11. Ans: No Suppose f : Z Z has the rule f(n) = 3n 1. Determine whether f is onto Z. Ans: No Suppose f : N N has the rule f(n) = 3n 2 1. Determine whether f is 11. Ans: Yes Suppose f : N N has the rule f(n) = 4n Determine whether f is onto N. Ans: No. Page 32
12 109. Suppose f : R R where f(x) = x/2. (a) Draw the graph of f. (b) Is f 11? (c) Is f onto R? Ans: (a) (b) No. (c) No Suppose f : R R where f(x) = x/2. (a) If S = {x 1 x 6}, find f(s). (b) If T = {3,4,5}, find f 1 (T). Ans: (a) {0,1,2,3} (b) [6,12) Determine whether f is a function from the set of all bit strings to the set of integers if f(s) is the position of a 1 bit in the bit string S. Ans: No; there may be no 1 bits or more than one 1 bit Determine whether f is a function from the set of all bit strings to the set of integers if f(s) is the number of 0 bits in S. Ans: Yes Determine whether f is a function from the set of all bit strings to the set of integers if f(s) is the largest integer i such that the ith bit of S is 0 and f(s) = 1 when S is the empty string (the string with no bits). Ans: No; f not defined for the string of all 1's, for example S = Let f(x) = x 3 /3. Find f(s) if S is: (a) { 2, 1,0,1,2,3}. (b) {0,1,2,3,4,5}. (c) {1,5,7,11}. (d) {2,6,10,14}. Ans: (a) { 3, 1,0,2,9}. (b) {0,2,9,21,41}. (c) {0,41,114,443}. (d) {2,72,333,914}. Page 33
13 115. Suppose f : R Z where f(x) = 2x 1. (a) Draw the graph of f. (b) Is f 11? (Explain) (c) Is f onto Z? (Explain) Ans: (a) (b) No. (c) Yes Suppose f : R Z where f(x) = 2x 1. (a) If A = {x 1 x 4}, find f(a). (b) If B = {3,4,5,6,7}, find f(b). (c) If C = { 9, 8}, find f 1 (C). (d) If D = {0.4,0.5,0.6}, find f 1 (D). Ans: (a) {1,2,3,4,5,6,7}. (b) {5,7,9,11,13}. (c) ( 9/2, 7/2]. (d) Suppose g : R R where (a) Draw the graph of g. (b) Is g 11? (c) Is g onto R? x 1 gx ( ) = 2. Ans: (a) (b) No. (c) No. Page 34
14 x Suppose g : R R where gx ( ) = 2. (a) If S = {x 1 x 6}, find g(s). (b) If T = {2}, find g 1 (T). Ans: (a) {0,1,2}. (b) [5,7) Show that x = x. Ans: Let n = x, so that n 1 < x n. Multiplying by 1 yields n + 1 > x n, which means that n = x Prove or disprove: For all positive real numbers x and y, x y x y. : x = y = Prove or disprove: For all positive real numbers x and y, x y x y. : x x, y y ; therefore xy x y ; since x y least as great as xy, then xy x y. is an integer at 122. Suppose g : A B and f : B C where A = {1,2,3,4}, B = {a,b,c}, C = {2,7,10}, and f and g are defined by g = {(1,b),(2,a),(3,a),(4,b)} and f = {(a,10),(b,7),(c,2)}. Find f g. Ans: {(1,7),(2,10),(3,10),(4,7)} Suppose g : A B and f : B C where A = {1,2,3,4}, B = {a,b,c}, C = {2,7,10}, and f and g are defined by g = {(1,b),(2,a),(3,a),(4,b)} and f = {(a,10),(b,7),(c,2)}. Find f 1. Ans: {(2,c),(7,b),(10,a)}. Use the following to answer questions : In the questions below suppose g : A B and f : B C where A = B = C = {1,2,3,4}, g = {(1,4),(2,1),(3,1),(4,2)} and f = {(1,3),(2,2),(3,4),(4,2)} Find f g. Ans: {(1,2),(2,3),(3,3),(4,2)} Find g f. Ans: {(1,1),(2,1),(3,2),(4,1)} Find g g. Ans: {(1,2),(2,4),(3,4),(4,1)}. Page 35
15 127. Find g (g g). Ans: {(1,1),(2,2),(3,2),(4,4)}. Use the following to answer questions : In the questions below suppose g : A B and f : B C where A = {1,2,3,4}, B = {a,b,c}, C = {2,8,10}, and g and f are defined by g = {(1,b),(2,a),(3,b),(4,a)} and f = {(a,8),(b,10),(c,2)} Find f g. Ans: {(1,10),(2,8),(3,10),(4,8)} Find f 1. Ans: {(2,c),(8,a),(10,b)} Find f f 1. Ans: {(2,2),(8,8),(10,10)} Explain why g 1 is not a function. Ans: g 1 (a) is equal to both 2 and 4. Use the following to answer questions : In the questions below suppose g : A B and f : B C where A = {a,b,c,d}, B = {1,2,3}, C = {2,3,6,8}, and g and f are defined by g = {(a,2),(b,1),(c,3),(d,2)} and f = {(1,8),(2,3),(3,2)} Find f g. Ans: {(a,3),(b,8),(c,2),(d,3)} Find f 1. Ans: {(2,3),(3,2),(8,1)} For any function f : A B, define a new function g : P(A) P(B) as follows: for every S A, g(s) = {f(x) x S}. Prove that f is onto if and only if g is onto. Ans: Suppose f is onto. Let T P(B) and let S = {x A f(x) T}. Then g(s) = T, and g is onto. If f is not onto B, let y B f(a). Then there is no subset S of A such that g(s) = {y}. Use the following to answer questions : In the questions below find the inverse of the function f or else explain why the function has no inverse. Page 36
16 135. f : Z Z where f(x) = x mod 10. Ans: f 1 (10) does not exist f : A B where A = {a,b,c}, B = {1,2,3} and f = {(a,2),(b,1),(c,3)}. Ans: {(1,b),(2,a),(3,c)} f : R R where f(x) = 3x x Ans: f ( x ) = f : R R where f(x) = 2x. 1 1 Ans: f () does not exist. 2 { x  2 if x f : Z Z where f(x) = x + 1 if x 4. Ans: f 1 (5) is not a single value Suppose g : A B and f : B C, where f g is 11 and g is 11. Must f be 11? Ans: No Suppose g : A B and f : B C, where f g is 11 and f is 11. Must g be 11? Ans: Yes Suppose f : R R and g : R R where g(x) = 2x + 1 and g f(x) = 2x Find the rule for f. Ans: f(x) = x + 5. Use the following to answer questions : In the questions below for each partial function, determine its domain, codomain, domain of definition, set of values for which it is undefined or if it is a total function: 143. f : Z R where f(n) = 1/n. Ans: Z, R, Z {0}, {0} f : Z Z where f(n) = n/2. Ans: Z, Z, Z, total function f : Z Z Q where f(m,n) = m/n. Ans: Z Z, Q, Z (Z {0}), Z {0}. Page 37
17 146. f : Z Z Z where f(m,n) = mn. Ans: Z Z, Z, Z Z, total function f : Z Z Z where f(m,n) = m n if m > n. Ans: Z Z, Z, {(m,n) m > n}, {(m,n) m n} For the partial function f : Z Z R defined by f( m, n) =, determine its 2 2 n m domain, codomain, domain of definition, and set of values for which it is undefined or whether it is a total function. Ans: Z Z, R, {(m,n) m n or m n}, {(m,n) m = n or m = n} Let f : {1,2,3,4,5} {1,2,3,4,5,6} be a function. (a) How many total functions are there? (b) How many of these functions are onetoone? Ans: (a) 6 5 = 7,776. (b) = 720. Use the following to answer questions : In the questions below find a formula that generates the following sequence a 1,a 2,a ,9,13,17,21,. Ans: a n = 4n ,3,3,3,3,. Ans: a n = ,20,25,30,35,. Ans: a n = 5(n + 2) ,0.9,0.8,0.7,0.6,. Ans: a n = 1 (n 1)/ ,1/3,1/5,1/7,1/9,. Ans: a n = 1/(2n 1) ,0,2,0,2,0,2,... Ans: a n = 1 + ( 1) n ,2,0,2,0,2,0,... Ans: a n = 1 + ( 1) n. Page 38
18 157. Find the sum 1/4 + 1/8 + 1/16 + 1/ Ans: 1/ Find the sum Ans: Find the sum Ans: + (2 ) Find the sum 1 1/2 + 1/4 1/8 + 1/16... Ans: 2/ Find the sum 2 + 1/2 + 1/8 + 1/32 + 1/ Ans: 8/ Find the summ Ans: 220, Find 164. Find i= 1 Ans: i i (( 2) 2 ). j j= 1 i= 1 Ans: 25. ij Rewrite ( i + 1) so that the index of summation has lower limit 0 and upper limit 7. 7 i= 0 2 Ans: ( i 3) + 1). 4 i= 3 Page 39
Chapter 3. if 2 a i then location: = i. Page 40
Chapter 3 1. Describe an algorithm that takes a list of n integers a 1,a 2,,a n and finds the number of integers each greater than five in the list. Ans: procedure greaterthanfive(a 1,,a n : integers)
More informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More information4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.
Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,
More information2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair
2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xyplane
More informationBasic Set Theory. Chapter Set Theory. can be written: A set is a Many that allows itself to be thought of as a One.
Chapter Basic Set Theory A set is a Many that allows itself to be thought of as a One.  Georg Cantor This chapter introduces set theory, mathematical induction, and formalizes the notion of mathematical
More informationDiscrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.
Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square
More informationApplications of Methods of Proof
CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The settheoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) :(a A) and (b B)}. The following points are worth special
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationAutomata and Formal Languages
Automata and Formal Languages Winter 20092010 Yacov HelOr 1 What this course is all about This course is about mathematical models of computation We ll study different machine models (finite automata,
More informationWe give a basic overview of the mathematical background required for this course.
1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The
More informationClicker Question. Theorems/Proofs and Computational Problems/Algorithms MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES
MC215: MATHEMATICAL REASONING AND DISCRETE STRUCTURES Tuesday, 1/21/14 General course Information Sets Reading: [J] 1.1 Optional: [H] 1.11.7 Exercises: Do before next class; not to hand in [J] pp. 1214:
More informationMatrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
More informationA Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions
A Second Course in Mathematics Concepts for Elementary Teachers: Theory, Problems, and Solutions Marcel B. Finan Arkansas Tech University c All Rights Reserved First Draft February 8, 2006 1 Contents 25
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationDefinition 14 A set is an unordered collection of elements or objects.
Chapter 4 Set Theory Definition 14 A set is an unordered collection of elements or objects. Primitive Notation EXAMPLE {1, 2, 3} is a set containing 3 elements: 1, 2, and 3. EXAMPLE {1, 2, 3} = {3, 2,
More information2.1 Sets, power sets. Cartesian Products.
Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects.  used to group objects together,  often the objects with similar properties This description of a set (without
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationSections 2.1, 2.2 and 2.4
SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete
More informationGeometric Transformations
Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted
More informationBoolean Algebra Part 1
Boolean Algebra Part 1 Page 1 Boolean Algebra Objectives Understand Basic Boolean Algebra Relate Boolean Algebra to Logic Networks Prove Laws using Truth Tables Understand and Use First Basic Theorems
More informationa 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
More informationYou know from calculus that functions play a fundamental role in mathematics.
CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.
More informationBaltic Way 1995. Västerås (Sweden), November 12, 1995. Problems and solutions
Baltic Way 995 Västerås (Sweden), November, 995 Problems and solutions. Find all triples (x, y, z) of positive integers satisfying the system of equations { x = (y + z) x 6 = y 6 + z 6 + 3(y + z ). Solution.
More informationElementary Number Theory and Methods of Proof. CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.
Elementary Number Theory and Methods of Proof CSE 215, Foundations of Computer Science Stony Brook University http://www.cs.stonybrook.edu/~cse215 1 Number theory Properties: 2 Properties of integers (whole
More informationAnswer Key for California State Standards: Algebra I
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
More informationDiscrete Mathematics Problems
Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 Email: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................
More informationFactoring Polynomials
Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent
More informationVocabulary Words and Definitions for Algebra
Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms
More informationProblem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00
18.781 Problem Set 7  Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list
More informationLecture Notes on Discrete Mathematics
Lecture Notes on Discrete Mathematics A. K. Lal September 26, 2012 2 Contents 1 Preliminaries 5 1.1 Basic Set Theory.................................... 5 1.2 Properties of Integers.................................
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More information6 Commutators and the derived series. [x,y] = xyx 1 y 1.
6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
More informationSets and set operations
CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used
More information2.1. Inductive Reasoning EXAMPLE A
CONDENSED LESSON 2.1 Inductive Reasoning In this lesson you will Learn how inductive reasoning is used in science and mathematics Use inductive reasoning to make conjectures about sequences of numbers
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More information4. An isosceles triangle has two sides of length 10 and one of length 12. What is its area?
1 1 2 + 1 3 + 1 5 = 2 The sum of three numbers is 17 The first is 2 times the second The third is 5 more than the second What is the value of the largest of the three numbers? 3 A chemist has 100 cc of
More informationSolutions Manual for How to Read and Do Proofs
Solutions Manual for How to Read and Do Proofs An Introduction to Mathematical Thought Processes Sixth Edition Daniel Solow Department of Operations Weatherhead School of Management Case Western Reserve
More information5.1 Commutative rings; Integral Domains
5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following
More informationCHAPTER 5. Number Theory. 1. Integers and Division. Discussion
CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a
More informationProperties of Real Numbers
16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should
More informationAlgebra I Vocabulary Cards
Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression
More information4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION
4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More information1 The Concept of a Mapping
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 1 The Concept of a Mapping The concept of a mapping (aka function) is important throughout mathematics. We have been dealing
More informationMathematics Review for MS Finance Students
Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,
More informationCardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.
Section 2.5 Cardinality (another) Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a bijection from A to B. If there is an injection
More information3. Equivalence Relations. Discussion
3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,
More informationBasic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011
Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely
More informationProblems on Discrete Mathematics 1
Problems on Discrete Mathematics 1 ChungChih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More information1. Determine all real numbers a, b, c, d that satisfy the following system of equations.
altic Way 1999 Reykjavík, November 6, 1999 Problems 1. etermine all real numbers a, b, c, d that satisfy the following system of equations. abc + ab + bc + ca + a + b + c = 1 bcd + bc + cd + db + b + c
More informationSAT Math Facts & Formulas Review Quiz
Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions
More informationRegular Languages and Finite State Machines
Regular Languages and Finite State Machines Plan for the Day: Mathematical preliminaries  some review One application formal definition of finite automata Examples 1 Sets A set is an unordered collection
More informationFoundations of Geometry 1: Points, Lines, Segments, Angles
Chapter 3 Foundations of Geometry 1: Points, Lines, Segments, Angles 3.1 An Introduction to Proof Syllogism: The abstract form is: 1. All A is B. 2. X is A 3. X is B Example: Let s think about an example.
More informationAlgebra 2 PreAP. Name Period
Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing
More informationINCIDENCEBETWEENNESS GEOMETRY
INCIDENCEBETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationChapter 7. Functions and onto. 7.1 Functions
Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiers
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationFlorida State University Course Notes MAD 2104 Discrete Mathematics I
Florida State University Course Notes MAD 2104 Discrete Mathematics I Florida State University Tallahassee, Florida 323064510 Copyright c 2011 Florida State University Written by Dr. John Bryant and Dr.
More informationFactoring Trinomials: The ac Method
6.7 Factoring Trinomials: The ac Method 6.7 OBJECTIVES 1. Use the ac test to determine whether a trinomial is factorable over the integers 2. Use the results of the ac test to factor a trinomial 3. For
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationPART I. THE REAL NUMBERS
PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS
More information+ Section 6.2 and 6.3
Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationMathematical induction. Niloufar Shafiei
Mathematical induction Niloufar Shafiei Mathematical induction Mathematical induction is an extremely important proof technique. Mathematical induction can be used to prove results about complexity of
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationA. The answer as per this document is No, it cannot exist keeping all distances rational.
Rational Distance Conor.williams@gmail.com www.unsolvedproblems.org: Q. Given a unit square, can you find any point in the same plane, either inside or outside the square, that is a rational distance from
More informationSolutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory
Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013
More informationUsing the ac Method to Factor
4.6 Using the ac Method to Factor 4.6 OBJECTIVES 1. Use the ac test to determine factorability 2. Use the results of the ac test 3. Completely factor a trinomial In Sections 4.2 and 4.3 we used the trialanderror
More informationAutomata Theory. Şubat 2006 Tuğrul Yılmaz Ankara Üniversitesi
Automata Theory Automata theory is the study of abstract computing devices. A. M. Turing studied an abstract machine that had all the capabilities of today s computers. Turing s goal was to describe the
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According
More informationElements of Abstract Group Theory
Chapter 2 Elements of Abstract Group Theory Mathematics is a game played according to certain simple rules with meaningless marks on paper. David Hilbert The importance of symmetry in physics, and for
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationDERIVATIVES AS MATRICES; CHAIN RULE
DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Realvalued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we
More informationThis chapter is all about cardinality of sets. At first this looks like a
CHAPTER Cardinality of Sets This chapter is all about cardinality of sets At first this looks like a very simple concept To find the cardinality of a set, just count its elements If A = { a, b, c, d },
More information1 = (a 0 + b 0 α) 2 + + (a m 1 + b m 1 α) 2. for certain elements a 0,..., a m 1, b 0,..., b m 1 of F. Multiplying out, we obtain
Notes on realclosed fields These notes develop the algebraic background needed to understand the model theory of realclosed fields. To understand these notes, a standard graduate course in algebra is
More information26 Integers: Multiplication, Division, and Order
26 Integers: Multiplication, Division, and Order Integer multiplication and division are extensions of whole number multiplication and division. In multiplying and dividing integers, the one new issue
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationQuotient Rings and Field Extensions
Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.
More informationDeterministic Finite Automata
1 Deterministic Finite Automata Definition: A deterministic finite automaton (DFA) consists of 1. a finite set of states (often denoted Q) 2. a finite set Σ of symbols (alphabet) 3. a transition function
More informationAnnouncements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets
CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.32.6 Homework 2 due Tuesday Recitation 3 on Friday
More informationFactoring Polynomials
UNIT 11 Factoring Polynomials You can use polynomials to describe framing for art. 396 Unit 11 factoring polynomials A polynomial is an expression that has variables that represent numbers. A number can
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationFactoring Quadratic Expressions
Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the
More informationDiscrete Mathematics
Discrete Mathematics ChihWei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can
More information7 Relations and Functions
7 Relations and Functions In this section, we introduce the concept of relations and functions. Relations A relation R from a set A to a set B is a set of ordered pairs (a, b), where a is a member of A,
More informationLinear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
More informationABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS
ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair
More informationChapter Two. Deductive Reasoning
Chapter Two Deductive Reasoning Objectives A. Use the terms defined in the chapter correctly. B. Properly use and interpret the symbols for the terms and concepts in this chapter. C. Appropriately apply
More informationElementary Number Theory We begin with a bit of elementary number theory, which is concerned
CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationH/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationPractice Book. Practice. Practice Book
Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Grade 10 Exam CAPS Grade 10 MATHEMATICS PRACTICE TEST ONE Marks: 50 1. Fred reads at 300 words per minute. The book he is reading has an average of
More informationBasic Concepts of Set Theory, Functions and Relations
March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory...1 1.1. Sets and elements...1 1.2. Specification of sets...2 1.3. Identity and cardinality...3
More informationArkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers
More informationUnited States Naval Academy Electrical and Computer Engineering Department. EC262 Exam 1
United States Naval Academy Electrical and Computer Engineering Department EC262 Exam 29 September 2. Do a page check now. You should have pages (cover & questions). 2. Read all problems in their entirety.
More information