Student Number: ____________________ Course: _____________ Maths Workbook 09/10 Table of Contents Table of Contents................................................................................................................1 Section 1: Sets.....................................................................................................................2 Introduction.................................................................................................................2 Standard Sets...............................................................................................................3 Predicates....................................................................................................................3 Set Operations.............................................................................................................4 Solving Problems Using Venn Diagrams...................................................................4 The Principle of Duality..............................................................................................6 Algebra of Sets............................................................................................................6 Further Properties of Sets............................................................................................7 Section 2: Relations............................................................................................................7 Binary Relations..........................................................................................................7 Equivalence Relations.................................................................................................9 Partial Orders..............................................................................................................9 Partitions...................................................................................................................10 Section 3: Functions..........................................................................................................10 Inverse Relations.......................................................................................................10 Composition of Relations.........................................................................................10 Functions...................................................................................................................12 Injective Surjective and Bijective Functions............................................................12 Inverse Functions......................................................................................................13 Composition of Functions.........................................................................................13 Pigeonhole Principle.................................................................................................13 Section 4: Logic................................................................................................................14 Propositions and Logic.............................................................................................14 Truth Tables..............................................................................................................14 Contrapositive Propositions......................................................................................15 Tautology..................................................................................................................15 Predicates and Quantifiers........................................................................................15 Disjunctive Normal Form.........................................................................................16 Logic Circuits............................................................................................................16 Section 5: Calculus...........................................................................................................17 Differentiation...........................................................................................................17 Product, Quotient and Chain Rules...........................................................................17 Slope of a Tangent to a Curve...................................................................................18 Turning Points...........................................................................................................18 Rates of Change........................................................................................................19 Section 6: Descriptive Statistics........................................................................................19 Types of Data............................................................................................................19 Frequency Distribution and Relative Frequency Distribution..................................19 Descriptive Measures................................................................................................20 Variance and Standard Deviation.............................................................................21 Section 7: Permutations and Combinations......................................................................21 Permutations.............................................................................................................21 Permutations with Repetition....................................................................................21 1Maths Workbook 09/10 Combinations............................................................................................................21 Combinations with Multiple Sets..............................................................................21 Section 8: Probability........................................................................................................22 Section 9: Inferential Statistics.........................................................................................23 Probability Distributions...........................................................................................23 Random Variables.....................................................................................................24 Mean Variance and Standard Deviation...................................................................24 The Normal Distribution...........................................................................................24 Appendix: The Normal Distribution Table.......................................................................26 Section 1: Sets Introduction A = {2,3,6,7} B = {orange, red, blue, green} For the sets A and B state whether each of the following is true or false a. 6 ∈ A b. Blue ∈ B 2Maths Workbook 09/10 c. Which of the following is true? • {2,3,4,5} = {5,4,3,2} • {5,6,7,8} = {6,7,8} • 6 ∈ {5,6,7,9} • 8 ∈ {5,6,7,9} d. Which of the following sets is equal to {6,7,8} • {6,7,8} • {7,6,8} • {7,8} • {8,7,6} Standard Sets e. Which of the following are true? • {-2,-1,0,1,2}∈ N • 0 ∈ N • 0 ∈ R • {1,2,3,4,5} ∈ Z Predicates a. List the elements of the following sets • {x:x ∈ z and x is an even number between 10 and 20} • {x:x ∈ z and x is an odd number} • {x|x = 2y and y {1,2,4,6}} • {x|x is an integer with 5 ≤ x <8} b. Write the following using suitable predicates • {January, June, July} • {Saturday, Sunday} • {2, 4, 6, 8, 10} • {a, e, i, o u} 3Maths Workbook 09/10 Set Operations a. A = {x|x ∈ N and 10 < x ≤ 20} B = {11,16,20} C = {x|x ∈ Z} D={11,12,13,14,15,16,17,18,19,20} Which of the following are true? • B ⊂ A • B ⊆ A • A ⊂ D • A ⊆ D • 12 ∈ A • {12} ⊆ A • {12} ∈ D b. U = {1,2,3,4,5,6,7,8,9,21, 22,23,24,25,26} A = {1,2,3,8,9} B = {8,1,3,25,26} C = {8,25,26,4} List the elements of the following sets: • A ∪ B • A ∩ B • A – B • ~ A • (A ∩ B) ∩ C c. Evaluate the following • {3,4,9} ∪ {5,8,11} • {3,4,9} ∪ {5,8,9} • {3,4,9} ∩ {5,8,9} • {red, green, blue} ∩ {blue, yellow} Solving Problems Using Venn Diagrams Solve the following problems using Venn Diagrams: a. In a group of 40 students, 28 study French and 20 study Commerce. If 13 students study both subjects, how many study neither? b. In a group of 30 students, 12 played tennis and 15 played football. If 8 played neither sport, how many played both? c. In a survey 100 people were asked which of three newspapers A, B and C they had read on the previous day. It was found that 29 people read A, 40 people read B and 38 people read C. Also, 10 people read A and B but not C. 12 people read B and C but not A. 7 people read A and C but not B and 30 people did not read any of the three newspapers. Find the number of people who read all three papers. 4Maths Workbook 09/10 Workspace: A B 5Maths Workbook 09/10 C The Principle of Duality Using an example, explain the principle of duality. Algebra of Sets a. Give the name for each of the following laws: • A ∪ (B ∪ C) = (A ∪ B) ∪ C • ~ (A ∪ B) = ~ A ∩ ~ B • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) • A ∪ ~ A = U • ~ U = Ø b. Let A={1,2,3,4,5} B={2,4,6,8} C={1,3,5,7,9} and U={1,2,3,4,5,6,7,8,9,10} Verify each of the following laws: • A ∪ A = A • A ∪ B = B ∪ A • ~ (A ∪ B) = ~ A ∩ ~ B • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) 6Maths Workbook 09/10 Further Properties of Sets c. Give the cardinality of the following sets: • {3,45,6,9} • {-1,-4,9,10,11,12} • {orange, blue} • {} • {0} d. Where A = {1,3,5,7,9} B = {2,4,6,8,10} and C = {1,2,3,4,5,6,7,8,9,10} Find: a. |A| b. |B| c. |C| d. |A ∪ B| e. |A ∪ C| f. |B ∪ C| e. Let A = {1,8} and B = {3,4} Find a. A X B Section 2: Relations Binary Relations 1. Write down the ordered pairs belonging to the following binary relations between A = {1,3,5,7} and B = {2,4,6} a. R = {(x,y): x+y = 9} b. S = {(x,y):x R given by f(x) = x2 and the function g: R -> R given by g(x) = 4x + 3. Calculate: i) g o f ii) f o g iii) f o f Pigeonhole Principle 8. There are 30 people in a class, show that at least two of them have a birthday in the same month. Workspace: 9. There are 79 two-child families in a certain village. Show that there are two families such that the months in which the birthdays of their two children fall are the same for each family. Workspace: 13Maths Workbook 09/10 Section 4: Logic Propositions and Logic 1. Let P, Q and R be propositions defined as follows: P: I am thirsty Q: My glass is empty R: It is three o clock Write each of the following propositions as logical expressions involving P, Q and R: a) I am thirsty and my glass is not empty b) It is three o clock and I am thirsty c) If it is three o clock, then I am thirsty d) If I am thirsty, then my glass is empty e) If I am not thirsty, then my glass is not empty Truth Tables 2. Draw Truth tables for each of the following: a) P and Q b) P and (not Q) c) Q then R d) (not P) then Q Workspace: 14Maths Workbook 09/10 Contrapositive Propositions 3. Show that (P => Q) => R is logically equivalent to (((not P) => R and (Q => R)) Workspace: Tautology 4. A compound proposition which is always true is called a tautology. By constructing truth tables, decide which of the following are tautologies: a. P or (not(P and Q)) b. not (P and Q) c. (P or Q) => R Workspace: Predicates and Quantifiers 5. Let x stand for cat and P(x) be the predicate x has whiskers. Write each of the following propositions in symbolic form. a) All cats have whiskers b) There is a cat with no whiskers c) No cat has whiskers d) Express the negation of (b) 15Maths Workbook 09/10 in symbol form e) Express the negation of (c) both in symbols and in English. Disjunctive Normal Form 6. The truth table for the Boolean function g = g(p, q, r, s) is shown below: a. Identify the corresponding minterms for this function b. Determine the disjunctive normal form p q r s g 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 0 1 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 1 1 1 1 0 0 0 1 1 0 1 1 1 1 1 0 0 1 1 1 1 0 Logic Circuits 7. Complete the table opposite which illustrates the inputs and outputs to each of the logic gates in the circuit shown in the diagram below. Gate Inputs Output 1 p, q pq 2 3 4 pq’, r pq’r 5 pq, r’ pqr’ 6 pqr v pq’r 7 pqr v pq’r, pqr’ 16Maths Workbook 09/10 Section 5: Calculus Differentiation 1. Differentiate the following equation with respect to x from first principles and then using the general rule: a. 2x2 – 7 b. 2x2 – 7 Product, Quotient and Chain Rules 2. Find dy/dx when: a. y = (x2 – 3x) (4x2 – 9) b. f(x) = 5x x3 + 2 c. f(x) = (5 – 2x) 4 17Maths Workbook 09/10 Workspace: Slope of a Tangent to a Curve 3. find the slope and the equation of a tangent to the curve y = 3x2 – 4x + 6 at the point (2,4) Turning Points 4. Find the turning point of the following quadratic function and state whether it is a min or max turning point: y = 2x2 – 4x + 8 18Maths Workbook 09/10 Rates of Change 5. The speed v in metres per second of a body after t seconds is given by: r = 3t(4 – t) a. Find the acceleration at each of the two instants when the speed is 9 metres per second b. Find the speed at the instant when the acceleration is zero Section 6: Descriptive Statistics Types of Data 1. Classify each of the following data sets as discrete or continuous data: a) The number of suitcases lost by an airline b) The height of corn plants c) The number of ears of corn produced by a corn plant d) The number of green m & m’s in a bag e) The time it takes for a car battery to die Frequency Distribution and Relative Frequency Distribution 2. The following table is a frequency distribution of the number of students studying different courses college. Course Number of Students Software Systems 180 Business Information Systems 250 Human Resource Management 270 Accounting 300 Total 1000 a) Find the relative frequency of this distribution b) Represent this as a percentage of the total number of students. 19Maths Workbook 09/10 Workspace: Descriptive Measures 3. Find the mean, median and mode values for each of the following sets of numbers: a. 1,2,3,4,5,6,7,8,9 b. 1,2,3,4,5,6,7,8,9,10 c. -500, 350, 475, -300, -500, 450, 425, -400 4. Below are the marks of a class tutorial, find the mean mark of the class Marks Number of Students 0 3 1 2 2 5 3 5 Total 15 5. Below are the scores of 50 teams in a table quiz. Find the mean score of the quiz. Score Frequency 0 -20 0 21 -30 2 31 -40 5 41 -50 7 51 -60 9 61 -70 20 71 -80 3 81 -90 3 91 -100 1 Total 50 20Maths Workbook 09/10 Variance and Standard Deviation 6. The following table is a distribution of the number of minutes that students spend talking on the phone per week. Number of Minutes Number of Students 0 – 4 6 5 – 9 16 10 – 14 13 15 – 19 12 20 – 25 3 Find the mean, variance and standard deviation of this distribution Section 7: Permutations and Combinations Permutations 1. Find the number of three letter words that can be formed from the word COMPUTER Permutations with Repetition 2. Find the number of eight letter words that can be formed from the word HARDWARE Combinations In How many ways can a team of 4 students be formed from a class of 20 students Workspace: Combinations with Multiple Sets 3. In how many ways can a sports day team consisting of 3 men and 4 women be chosen from 7 men and 5 women? 21Maths Workbook 09/10 Workspace: Section 8: Probability 1. A coin is weighted so that tails is twice as likely to appear as heads. Find the P(T) and P(H). 2. Two dogs d1 and d2 and three cats c1, c2 and c3 are in a race. Those of the same species have equal probabilities of winning, but each dog is twice as likely to win as any cat. i) Find the probability that a cat wins the race ii) If one of the dogs and one of the cats is called Daisy, find the probability that an animal called Daisy will win the race. 3. Three batteries are chosen at random from 15 batteries of which 5 are defective. Find the probability p that: i) None are defective ii) Exactly one is defective iii) At least one is defective 22Maths Workbook 09/10 4. Six pairs of twins are standing in a room, where each pair consists of a male and a female. i) If two people are chosen at random, find the probability that a) They are brother and sister b) One is male and one is female ii) If 4 people are chosen at random, find the probability that a) 2 pairs of twins are chosen b) No pair of twins are among the chosen 4 Section 9: Inferential Statistics Probability Distributions 1. Suppose a student is sitting 4 exams and is interested in the number of pass and fail results that he/she might get. Given that the probability of a pass result is .75 and the probability of a fail result is .25. Find the probability that the student will get: a. All Passes b. All Fails c. 2 passes and 2 fails d. 3 fails and 1 pass e. 3 passes and 1 fail 23Maths Workbook 09/10 Random Variables 2. Suppose that the student was only interested in how many modules he/she might pass. a. Using a random variable, provide the probability distribution for how many modules the student might pass. b. What is the probability of passing less than 3 exams? c. What is the probability of passing all 4 modules? d. What is the probability of passing less than 1 module? Mean Variance and Standard Deviation 3. For the probability distribution in question 2a find: a. Mean b. Variance c. Standard Deviation The Normal Distribution 4. If z has a standard normal distribution, using the normal distribution table find: a. P(z ≤ 1.25) b. P(z > 1.25) c. P(-1.25 ≤ z ≤ 1.25) d. P(z ≤ 1.00) e. P(z > 1.00) f. P(-1.00 ≤ z ≤ 1.00) g. P(z ≤ 3.6) h. P(z > 3.6) i. P(-3.6≤ z ≤ 3.6) j. P(z ≤ 3.99) 5. A random variable X is normally distributed with mean 100 and standard deviation 10. Find the following probabilities: a. P(X > 85) 24Maths Workbook 09/10 b. P(X < 115) c. P(X > 130) d. P(95 < X < 105) 6. If a test is normally distributed with mean 50 and standard deviation 10. What is the proportion of students scoring between 60 and 80 in the test? 7. A group of students took an IQ test with a mean of 100 and a standard deviation of 10. a. What percentage of students should score between 1 standard deviation above (100 + 10) the mean and 1 standard deviation below (100 -10) the mean? b. In this same test, one student received a score of 130. How did this score rate against the scores of the other students? 25Maths Workbook 09/10 26 Appendix: The Normal Distribution Table Math’s Workbook – Grading Rubric Criteria 4 marks 3 marks 2 marks 1 mark 0 marks Completeness All tasks have been attempted with fully worked solutions Most tasks have been attempted with fully worked solutions Most tasks have been attempted but solutions are incomplete or calculations are not shown Few tasks have been attempted with incomplete solutions and little or no evidence of calculation Little or no tasks have been completed with no evidence of calculation OR workbook is not presented for assessment Review Evidence of review, correction and where necessary, recalculation of incorrect solutions throughout all tasks Evidence of review, correction and where necessary, recalculation of incorrect solutions throughout most tasks Evidence of review and correction but little evidence of recalculation of incorrect solutions Little evidence of review correction or recalculation of incorrect answers No evidence of review, correction or recalculation of correct answers OR workbook is not presented for assessment Continuous Effort Evidence of regular effort in both lectures and tutorials over 13 weeks Evidence of: regular effort in either lectures or tutorials OR improved effort in both lectures and tutorials throughout the semester No evidence of regular effort OR workbook is not presented for assessment