Section 1.1 Inductive Reasoning
What You Will Learn Inductive and deductive reasoning processes 1.1-2
Natural Numbers The set of natural numbers is also called the set of counting numbers. N = {1, 2, 3, 4, 5, 6, 7, 8, } The three dots, called an ellipsis, mean that 8 is not the last number but that the numbers continue in the same manner. 1.1-3
Divisibility If a b has a remainder of zero, then a is divisible by b. The even counting numbers are divisible by 2. They are 2, 4, 6, 8,. The odd counting numbers are not divisible by 2. They are 1, 3, 5, 7,. 1.1-4
Inductive Reasoning The process of reasoning to a general conclusion through observations of specific cases. Also called induction. Often used by mathematicians and scientists to predict answers to complicated problems. 1.1-5
Example 3: Inductive Reasoning What reasoning process has led to the conclusion that no two people have the same fingerprints or DNA? This conclusion has resulted in the use of fingerprints and DNA in courts of law as evidence to convict persons of crimes. 1.1-6
Example 3: Inductive Reasoning Solution: In millions of tests, no two people have been found to have the same fingerprints or DNA. By induction, then, we believe that fingerprints and DNA provide a unique identification and can therefore be used in a court of law as evidence. Is it possible that sometime in the future two people will be found who do have exactly the same fingerprints or DNA? 1.1-7
Scientific Method Inductive reasoning is a part of the scientific method. When we make a prediction based on specific observations, it is called a hypothesis or conjecture. 1.1-8
Example 5: Pick a Number, Any Number Pick any number, multiply the number by 4, add 2 to the product, divide the sum by 2, and subtract 1 from the quotient. Repeat this procedure for several different numbers and then make a conjecture about the relationship between the original number and the final number. 1.1-9
Example 5: Pick a Number, Any Number Solution: Pick a number: say, 5 Multiply the number by 4: 4 5 = 20 Add 2 to the product: 20 + 2 = 22 Divide the sum by 2: 20 2 = 11 Subtract 1 from quotient: 11 1 = 10 1.1-10
Example 5: Pick a Number, Any Number Solution: We started with the number 5 and finished with the number 10. Start with the 2, you will end with 4. Start with 3, final result is 6. 4 would result in 8, and so on. We may conjecture that when you follow the given procedure, the number you end with will always be twice the original number. 1.1-11
Counterexample In testing a conjecture, if a special case is found that satisfies the conditions of the conjecture but produces a different result, that case is called a counterexample. Only one exception is necessary to prove a conjecture false. If a counterexample cannot be found, the conjecture is neither proven nor disproven. 1.1-12
Deductive Reasoning A second type of reasoning process is called deductive reasoning. Also called deduction. Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. 1.1-13
Example 6: Pick a Number, n Prove, using deductive reasoning, that the procedure in Example 5 will always result in twice the original number selected. Note that for any number n selected, the result is 2n, or twice the original number selected. 1.1-14
Example 6: Pick a Number, n Solution: To use deductive reasoning, we begin with the general case rather than specific examples. Pick a number: n Multiply the number by 4: 4n Add 2 to the product: 4n + 2 Divide the sum by 2: (4n + 2) 2 = 2n + 1 Subtract 1 from quotient: 2n + 1 1 = 2n 1.1-15
Section 2.1 Set Concepts 2.1-16
What You Will Learn Equality of sets Application of sets Infinite sets 2.1-17
Set A set is a collection of objects, which are called elements or members of the set. Three methods of indicating a set: Description Roster form Set-builder notation 2.1-18
Well-defined Set A set is well defined if its contents can be clearly defined. Example: The set of U.S. presidents is a well defined set. Its contents, the presidents, can be named. 2.1-19
Example 1: Description of Sets Write a description of the set containing the elements Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday. 2.1-20
Example 1: Description of Sets Solution The set is the days of the week. 2.1-21
Roster Form Listing the elements of a set inside a pair of braces, { }, is called roster form. Example {1, 2, 3,} is the notation for the set whose elements are 1, 2, and 3. Non-Examples (1, 2, 3,) and [1, 2, 3] 2.1-22
Naming of Sets Sets are generally named with capital letters. Definition: Natural Numbers The set of natural numbers or counting numbers is N. N = {1, 2, 3, 4, 5, } 2.1-23
Example 2: Roster Form of Sets Express the following in roster form. a) Set A is the set of natural numbers less than 6. Solution: a) A = {1, 2, 3, 4, 5} 2.1-24
Example 2: Roster Form of Sets Express the following in roster form. b) Set B is the set of natural numbers less than or equal to 80. Solution: b) B = {1, 2, 3, 4,, 80} 2.1-25
Example 2: Roster Form of Sets Express the following in roster form. c) Set P is the set of planets in Earth s solar system. Solution: c) P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} 2.1-26
Set Symbols The symbol, read is an element of, is used to indicate membership in a set. The symbol means is not an element of. 2.1-27
Set-Builder Notation (or Set-Generator Notation) A formal statement that describes the members of a set is written between the braces. A variable may represent any one of the members of the set. 2.1-28
Example 4: Using Set-Builder Notation a) Write set B = {1, 2, 3, 4, 5} in set-builder notation. b) Write in words, how you would read set B in set-builder notation. 2.1-29
Example 4: Using Set-Builder Notation Solution a) or B x x N and x 6 B x x N and x 5 b) The set of all x such that x is a natural number and x is less than 6. 2.1-30
Example 6: Set-Builder Notation to Roster Form Write set in roster form. A x x N and 2 x 8 Solution A = {2, 3, 4, 5, 6, 7} 2.1-31
Write set A using set builder notation. A 2,4,6,8,10,12,14 a. b. c. d. x x is an even counting number and 2 x 14 x x is an integer and 2 x 14 x x is an even counting number and 2 x 16 x x is an even counting number and 0 x 14
Express set A with a written description. a. Counting numbers between 2 and 14, inclusive b. Counting numbers between 2 and 14, exclusive c. Even counting numbers between 2 and 14, exclusive d. Even counting numbers between 2 and 14, inclusive A 2,4,6,8,10,12,14
Finite Set A set that contains no elements or the number of elements in the set is a natural number. Example: Set B = {2, 4, 6, 8, 10} is a finite set because the number of elements in the set is 5, and 5 is a natural number. 2.1-34
Determine whether the statement is true or false. a. True 7 1, 3, 5, 7 b. False c. Can t determine
Infinite Set A set that is not finite is said to be infinite. The set of counting numbers is an example of an infinite set. 2.1-36
Determine whether the statement is true or false. The set of even natural numbers greater than 100 is an infinite set. a. True b. False c. Can t determine
Equal Sets Set A is equal to set B, symbolized by A = B, if and only if set A and set B contain exactly the same members. Example: { 1, 2, 3 } = { 3, 1, 2 } 2.1-38
Determine whether the statement is true or false. a. True b. False c. Can t determine
Determine whether the statement is true or false. a. True rose, tulip, lily rose, lily, daisy b. False c. Can t determine
Cardinal Number The cardinal number of set A, symbolized n(a), is the number of elements in set A. Example: A = { 1, 2, 3 } and B = {England, Brazil, Japan} have cardinal number 3, n(a) = 3 and n(b) = 3 2.1-41
Equivalent Sets Set A is equivalent to set B if and only if n(a) = n(b). Example: D={ a, b, c }; E={apple, orange, pear} n(d) = n(e) = 3 So set A is equivalent to set B. 2.1-42
Equivalent Sets - Equal Sets Any sets that are equal must also be equivalent. Not all sets that are equivalent are equal. Example: D ={ a, b, c }; E ={apple, orange, pear} n(d) = n(e) = 3; so set A is equivalent to set B, but the sets are NOT equal 2.1-43
Determine whether the statement is true or false. a. True 1, 2, 3 is equivalent to 1, 2, 3, 4 b. False c. Can t determine
One-to-one Correspondence Set A and set B can be placed in oneto-one correspondence if every element of set A can be matched with exactly one element of set B and every element of set B can be matched with exactly one element of set A. 2.1-45
One-to-one Correspondence Consider set S states, and set C, state capitals. S = {North Carolina, Georgia, South Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} Two different one-to-one correspondences for sets S and C are: 2.1-46
One-to-one Correspondence S = {No Carolina, Georgia, So Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} S = {No Carolina, Georgia, So Carolina, Florida} C = {Columbia, Raleigh, Tallahassee, Atlanta} 2.1-47
One-to-one Correspondence Other one-to-one correspondences between sets S and C are possible. Do you know which capital goes with which state? 2.1-48
Null or Empty Set The set that contains no elements is called the empty set or null set and is symbolized by or. 2.1-49
Null or Empty Set Note that { } is not the empty set. This set contains the element and has a cardinality of 1. The set {0} is also not the empty set because it contains the element 0. It has a cardinality of 1. 2.1-50
Universal Set The universal set, symbolized by U, contains all of the elements for any specific discussion. When the universal set is given, only the elements in the universal set may be considered when working with the problem. 2.1-51
Universal Set Example If the universal set is defined as U = {1, 2, 3, 4,,,10}, then only the natural numbers 1 through 10 may be used in that problem. 2.1-52
Section 2.2 Subsets
What You Will Learn Subsets and proper subsets 2.2-54
Subsets Set A is a subset of set B, symbolized A B, if and only if all elements of set A are also elements of set B. The symbol A B indicates that set A is a subset of set B. 2.2-55
Subsets The symbol A B set A is not a subset of set B. To show that set A is not a subset of set B, one must find at least one element of set A that is not an element of set B. 2.2-56
Determining Subsets Example: Determine whether set A is a subset of set B. A = { 3, 5, 6, 8 } B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Solution: All of the elements of set A are contained in set B, so A B. 2.2-57
Determine whether the statement is true or false. a. True 5 x x N and x 5 b. False c. Can t determine
Proper Subset Set A is a proper subset of set B, symbolized A B, if and only if all of the elements of set A are elements of set B and set A B (that is, set B must contain at least one element not is set A).
Determining Proper Subsets Example: Determine whether set A is a proper subset of set B. A = { dog, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, and sets A and B are not equal, therefore A B. 2.2-60
Determining Proper Subsets Example: Determine whether set A is a proper subset of set B. A = { dog, bird, fish, cat } B = { dog, cat, bird, fish } Solution: All the elements of set A are contained in set B, but sets A and B are equal, therefore A B. 2.2-61
Determine whether the statement is true or false. a. True dog cat, dog, mouse b. False c. Can t determine
Number of Distinct Subsets The number of distinct subsets of a finite set A is 2 n, where n is the number of elements in set A. 2.2-63
Number of Distinct Subsets Example: Determine the number of distinct subsets for the given set {t, a, p, e}. List all the distinct subsets for the given set {t, a, p, e}. 2.2-64
Number of Distinct Subsets Solution: Since there are 4 elements in the given set, the number of distinct subsets is 2 4 = 2 2 2 2 = 16. {t,a,p,e}, {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { } 2.2-65
Determine whether the statement is true or false. a. True b, a, t has eight subsets b. False c. Can t determine
Number of Distinct Proper Subsets The number of distinct proper subsets of a finite set A is 2 n 1, where n is the number of elements in set A. 2.2-67
Number of Distinct Proper Subsets Example: Determine the number of distinct proper subsets for the given set {t, a, p, e}. 2.2-68
Number of Distinct Subsets Solution: The number of distinct proper subsets is 2 4 1= 2 2 2 2 1 = 15. They are {t,a,p}, {t,a,e}, {t,p,e}, {a,p,e},{t,a}, {t,p}, {t,e}, {a,p}, {a,e}, {p,e}, {t}, {a}, {p}, {e}, { }. Only {t,a,p,e}, is not a proper subset. 2.2-69