Inductive and Inductive Reasoning (1.1)

Transcription

1 1 Inductive and Inductive Reasoning (1.1) I. Inductive Reasoning The process of arriving at a general conclusion (hypothesis/conjecture) based on observations of specific values. There is no guarantee that the conclusions are true. Examples: a) Identify the pattern. Use this pattern to find the next number. 1, 1, 2, 3, 5, 8, 13, 21, Fibonacci Sequence 3, 6, 18, 36, 108, 216, b) Use inductive reasoning to predict the next line

2 2 c) Describe the two patterns in the sequence. Use the patterns to describe the next figure. d) The triangular arrangement of numbers shown below is known as Pascal s Triangle, credited to French mathematician Blaise Pascal. Use inductive reasoning to find the six numbers designated by question marks. (Blitzer, p 11 # 62) ?????? 2

3 3 Deductive Reasoning The process of proving a specific conclusion (theorem) from one or more general statements. Conclusions based on deductive reasoning are true. Steps for the examples below: 1. Select a few numbers and follow the steps. 2. Write a conjecture based on your examples. 3. Name a variable. 4. Use deductive reasoning to prove the conjecture true. Examples: a) Select a number. Add three. Multiply the sum by two. Subtract six from the product. Divide the result by two. Original number selected Result of the process b) Identify the reasoning process as induction or deduction. The course policy states that work turned in late will be marked down a grade. I turned in my report a day late, so it was marked down from B to C. 3

4 4 Basic Set Concepts (2.1) I. Set A collection of objects whose contents can be clearly determined. Capitol letters usually name a set. Elements are the contents in a set. Sets can be described using words, the roster method, or set-builder notation. Examples: a) Word description: A represents the set of 52 playing cards. b) Roster Method: 1. List all elements in the set inside a pair of { }. 2. Separate each element with a comma Rolling a die. B = {1, 2, 3, 4, 5,6 } Rolling a pair of dice. C = { (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)} Tossing a coin. D = {Heads, Tails} c) Set-Builder Notation: General format is { x x.}. E = {x x is the number of days in a month.} Set E is the set of all elements x such that x is the number of days in a month. F = { x x is a number less than 4 or greater than 10} G = { x x is a number less than 4 and greater than 10} 4

5 5 II. Empty Set/Null Set A set containing no elements because the events are impossible. Notation: {} or III. - indicates that an object is an element of a set. Read: is an element of Example: 3 { x x is a number less than 4 or greater than 10} IV. - indicates that an object is not an element of a set. Read: is not an element of Example: 4 { x x is a number less than 4 or greater than 10} V. Set of Natural Numbers N = {1, 2, 3, 4, } Examples: Express each of the following sets using the roster method. a) G is the set of natural numbers less than 5. b) H = {x x N and x is a multiple of 2x+1} Examples: Express each of the following sets using set-builder notation. c) I represents the set of natural numbers greater than or equal to 70 and less than or equal to 100. d) J represents the set of natural numbers greater than 70 and less than 100. VI. Review of Inequality Symbols. x < a {x x N and x < a} x a {x x N and x a} x > a {x x N and x > a} x a {x x N and x a} 5

6 6 VII. Cardinal Number of set A the number of distinct elements in set A. Notation: n(a) Example: Find the cardinal number for the sets below. a) J = {1, 2, 3, 4} b) K = {1, 2, 3, 4, 1} c) L = {0} d) M = {} Finite Set A set whose cardinality is 0 or a natural number. Infinite Set A set whose cardinality is not 0 or a natural number. VIII. Set A is the equivalent of set B set A and set B contain the same number of distinct elements. The definition is equivalent to saying n(a) = n(b). Example: Are the sets below equivalent? N = {Florida, Alabama, Georgia} O = {0.2, 4, 1.2} 5 Example: Are the sets below equivalent? P = {steak, chicken, pork, ribs} Q = {1, 2, 3, 1, 4} IX. Set A is equal to set B set A and Set B have exactly the same elements. Notation: A = B The order of the elements or repetition in elements does not matter. Example: Identify as True or False. { B, E, L, L, L, L, A} = {A, L, E, B} 6

7 7 SUBSETS (2.2) fill in exs I. Set A is a subset of set B every element in set A is also an element in set B. Notation: A B The null set is a subset of any set B. {} B. Sets A and B are equal if A B and B A Example: { 1, 2, 3} {1, 2, 3, 3} and { 1, 2, 3, 3} {1, 2, 3}, so { 1, 2, 3} { 1, 2, 3, 3} II. Set A is not a subset of set B There is at least one element of set A that is not an element of set B. Notation: A B *** The computer doesn t have the correct symbol here. Draw in a!! III. Set A is a proper subset of set B- Set A is a subset of set B and sets A and B are not equal. Notation: A B The null set is a proper subset of any set B, provided that B is not the null set. {} B. Example: { 1, 2, 3} {0, 1, 2, 3, 4, 5} Examples on element vs. set notation: Label each statement as true or false. a) 7 {x x N and x < 9 } b) {7} {x x N and x < 9 } c) 7 {x x N and x < 9 } d) {7} {x x N and x < 9 } 7

8 8 IV. Number of Subsets The number of subsets of a set with n elements is n 2. The number of proper subsets of a set with n elements is n 2-1. There are an infinite number of subsets for an infinite set. Examples: Calculate the number of subsets and the number of proper subsets of each set. Don t forget that the null set is a subset of any set! a) {x x is the month of the year} b) {x x N and x < 3} 8

9 9 Venn Diagrams and Set Operations with Two Sets (2.3) I. Universal Set set containing all the elements being considered in a problem. Notation: U. The set of all possible outcomes. The set of all sample points. Example: The set of all Delta domestic flights in The set of all H.C.C. left handed students. Venn Diagram II. Disjoint set Two sets with no elements in common. Venn Diagram Example {The next sale by a PC retailer is a laptop computer}, {The next sale by a PC retailer is a desktop computer} {42 right handed students} and {36 left handed students} III. Proper subset Venn Diagram Example: A = { x x N} B = {x x N and x is even} 9

10 10 IV. Equal sets Venn Diagram Example: A = {0, 2, 4, 6} B = {x x = 2n and 0 n 3} V. Compliment of set A set of all elements in the universal set that are not in A. Notation: A A = {x x U and x A}. Look for the word not Venn Diagram Example: Consider rolling a dice once. Define the compliment of each event. Find the compliment of each set. a) W= {A score less than four} = {1, 2, 3} W = b) Z = {A score that is even} = {2, 4, 6} Z = 10

11 11 VI. Intersection of sets A and B the set of elements common to both set A and set B. Notation: A B A B = {x x A and x B} Look for the word and. Look for the word but A {} = {} Venn Diagram Example: Let event A={2, 8, 14, 18} and event B={4, 6, 8, 10, 12}. What is the intersection of A and B? VII. Union of sets A and B the set of all events that are in A or B or both. Notation: A B A B = {x x A or x B} Look for the word or. A {} = A Venn Diagram Example: Let event A={2, 8, 14, 18} and event B={4, 6, 8, 10, 12}. What is the union of A and B? 11

12 12 Example A six-sided, fair die is rolled. Let event A = {2, 4, 6}, the die is even. Let event B = {3, 6}, the die is a multiple of three. a) What is A B? b) What is (A B)? c) What is (A ) B? 12

13 13 d) What is A A? e) What is A B C? 13

14 14 VIII. Cardinal Number of the Union of Two Finite Sets. n(a B) = n(a) + n(b) n(a B) Example: Set A contains 3 letters and 5 numbers, set B contains 7 letters and 2 numbers, and 1 letter and 2 numbers are common to both sets. How many elements are in set A or set B? 14

15 15 Venn Diagrams and Set Operations with Three Sets (2.4) Whole Numbers: {0, 1, 2, 3, } Example: Let U = {x x W and x 15} and A = {x x W and x 7}, B = {x x is odd and x 15}, and C = {x x is even and x 12}. Find the following: Steps: 1. Find the four intersections 2. Place elements into regions, starting with the innermost region and then work outwards a) ((C A) (B A)) b) (A B) C c) B (A C) 15

16 16 Venn Diagrams with Three Sets Ex. Use the Venn diagram to find which regions are represented by a) (A B) b) A B c) A B d) (A B) e) (A B) C 16

17 17 DeMorgan s Laws 1. (A B) = A B 2. (A B) = A B Deductive proof of (A B) = A B Set A Regions in the Venn Diagram B A B (A B) Set A Regions in the Venn Diagram B A B 17

18 18 Deductive proof of (A B) = A B Set A Regions in the Venn Diagram B A B (A B) Set A Regions in the Venn Diagram B A B Steps to Prove the Equality of Sets: 1. Create a Venn Diagram. 2. Identify the region defined by the left hand side of the equality in the Venn diagram. 3. Identify the region defined by the right hand side of the equality in the Venn diagram. 4. If both sets are identified by the same region, then the sets are equal. 18

19 19 Survey Problems (2.5) Steps to solve Survey Problems: 1. Use the problem description to define sets. 2. Draw a Venn diagram. 3. Label the cardinality of the innermost region of the Venn Diagram, which represents the intersection of the sets. 4. Working outwards, label the cardinality of the remaining regions. Example: A survey of 100 college students was taken to determine preferences in cheeseburger condiments. Forty-three use mayonnaise, 52 use mustard, and 35 use ketchup. Thirteen use mayonnaise and mustard, 12 use mayonnaise and ketchup, 14 use ketchup and mustard, and 5 use all three. a) How many students do not use any condiment? b) How many students use ketchup only? c) How many students use mayonnaise only? d) How many students use at least two of the condiments? e) How many students use exactly one condiment? f) How many students use at most two of the condiments? 19

20 20 Example A survey of 80 college students was taken to determine the musical styles they listened to. Forty-two students listened to rock, 34 to classical, and 27 to jazz. Twelve students listened to rock and jazz, 14 to rock and classical, and 10 to classical and jazz. Seven students listened to all three musical styles. Of those surveyed: a) How many listened to classical and jazz, but not rock? b) How many listened to music in exactly one of the musical styles? c) How many listened to music in at least two of the musical styles? d) How many did not listen to any of the musical styles? 20