2 Chapter Numeration Systems and Sets Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 2-2 Describing Sets In the years from 1871 through 1884, Georg Cantor created set theory which has a profound effect on research and mathematical teaching. Sets, and relations between sets, are a basis for teaching children the concept of a whole number and the concept of “less than” as well as addition, subtraction, and multiplication of whole numbers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. The Language of Sets A set is any group or collection of objects. The objects that belong to a set are the elements, or members, of the set. One method of denoting a set is to simply list the elements inside braces and label the set with a capital letter. A = {1, 2, 3, 4, 5} Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. The Language of Sets Each element of a set is listed only once. The order of the elements is unimportant. We symbolize that an element belongs to a set using the symbol , and we use to indicate that an element does not belong to a set. A set must be well-defined. We must be able to tell whether or not an object belongs to the set. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Listing Method (Roster Method) Methods of Describing Sets Listing Method (Roster Method) C = {red, green, yellow, blue} N = {1, 2, 3, 4, … } natural numbers L = {a, b, c, d, …} lower case letters Ellipsis – continues in the same manner Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Methods of Describing Sets Set-Builder Notation x is an element of the set of natural numbers and x is less than 11 the set of all elements x such that Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-6 Write the following sets using set-builder notation. a. {2, 4, 6, 8, 10, …} b. {1, 3, 5, 7, …} a. {x | x is an even natural number} or {x | x = 2n, n  N} b. {x | x is an odd natural number} or {x | x = 2n − 1, n  N} Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-7a Write the set A = {2k + 1 | k = 3, 4, 5} by listing its elements. A = {7, 9, 11} Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-7b Write the set B = {x|x is a positive even natural number less than 8} B = {2, 4, 6} Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Two sets are equal if, and only if, they contain the same elements. The order of the elements makes no difference. If set A is equal to set B, we write A = B. If A = B, then every element of set A is contained in set B, and every element of set B is contained in set A. If A ≠ B , then there is at least one element that is not contained in both sets A and B. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition One-to-One Correspondence If the elements of sets P and S can be paired so that for each element of P there is exactly one element of S and for each element of S there is exactly one element of P, then the two sets P and S are said to be in a one-to-one correspondence. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

One-to-One Correspondence Set {1, 2, 3} can be placed in a one-to-one correspondence with set {a, b, c} by establishing either of the following pairings: These are not the only possible pairs to establish a one-to-one correspondence. We would like to list all the possibilities. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

One-to-One Correspondence One way to illustrate all possible one-to-one correspondences is to use a table; another is a tree. 1 2 3 a b c 1 2 3 b c c b a a c c a b a b b a c 6 6 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. How can we determine the number of one-to-one correspondences between two sets? In our example, given the element 1 in the first set, there are three possible elements in the second set which can be paired with the 1, namely a, b, or c. Once we have paired an element with 1, that leaves two elements in the second set to pair with the number 2. Once we have paired an element in the second set with 1 and with 2, there will be one element left in the second set to pair with 3. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Fundamental Counting Principle If event M can occur in m ways and, after it has occurred, event N can occur in n ways, then event M followed by event N can occur in m • n ways. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Equivalent Sets Two sets A and B are equivalent, written A ~ B, if there exists a one-to-one correspondence between the sets. Recall if two sets are equal, they have the same elements. Equal sets are equivalent, but equivalent sets are not necessarily equal. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-8 Given sets A = {p, q, r, s} B = {a, b, c} C = {x, y, z} D = {b, a, c} determine whether the following statements are true or false. T F T F F F F T Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition The cardinal number of a set S, denoted n(S), indicates the number of elements in the set S. What is the cardinality of each of the following? K = {3, 5, 8, 0, 11} L = {0} X = {x | x is a real number and x = x + 1} Y = Ø n(K) = 5 n(L) = 1 n(X) = 0 n(Y) = 0 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Finite Sets vs. Infinite Sets A set is finite if its cardinality is zero or a natural number. Examples The set of presidents of the United States. The set of planets in our solar system. An infinite set is a set that is not finite. Examples The set of whole numbers = { 0, 1, 2, 3, …}. The set of solutions to the equation x + 1 = x + 1. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. More About Sets The universal set U is defined as that set consisting of all elements under consideration. We usually denote the universal set with a capital U. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Universal Set Suppose the universal set is defined as: U = {x | x is a member of the U.S. Senate} Denote the universal set with a large rectangle, and particular sets are indicated by geometric figures inside the rectangle. In this case, only members of the U.S. Senate are included in the rectangle. This pictorial representation is a Venn diagram. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Universal Set U Venn Diagram Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Universal Set Consider the members of the universal set, U, who are females. Since this set is wholly contained in U but does not contain all the members of U, we denote this set with a circle as follows: F = {x | x is a female U.S. senator} U F Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Complement of a Set Consider the members of the universal set, U, who are not females. This is the set of all members of the set, U, that are not in set F. We refer to this set as the complement of set F and denote it by the shaded region in the figure below. F = {x | x is a female U.S. senator} U F Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition The complement of a set F, written is the set of all elements in the universal set U that are not in F; that is, Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-9 a. If U = {a, b, c, d} and B = {c, d}, find b. If U = {x | x is an animal in the zoo} and S = {x | x is a snake in the zoo}, describe = {x | x is a zoo animal that is not a snake} Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-9 (continued) c. If U = N, E = {2, 4, 6, 8, …}, and O = {1, 3, 5, 7,…}, find Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Set B is a subset of set A, written if and only if every element of B is an element of A. When a set A is not a subset of another set B, we write Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Subset vs. Equality of Sets The definition allows for B to be equal to A. Underscore B is a subset of A if and only if every element of B is an element of A. This implies this This implies this Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Proper Subsets Suppose set B is a subset of set A, and B is not equal to A. U B A Then there must be at least one element of A that is not in B. We say that B is a proper subset of A, written Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-10 Given A = {1, 2, 3, 4, 5}, B = {1, 3}, P = {x | x = 2n − 1, where n N}. a. Which sets are subsets of each other? Because 21 − 1 = 1, 22 − 1 = 3, 23 − 1 = 7, 24 − 1 = 15, and so on, P = {1, 3, 7, 15, …}. So, Also Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-10 (continued) Given A = {1, 2, 3, 4, 5}, B = {1, 3}, P = {x | x = 2n − 1, where n N}. b. Which sets are proper subsets of each other? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Inequalities: An Application of Set Concepts Suppose A and B are finite sets. If A = B, then n(A) = n(B), and A and B are equivalent. It is possible to establish a one-to-one correspondence between A and B. If A is a proper subset of B, then n(A) < n(B), and it is not possible to establish a one-to-one correspondence between A and B. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Suppose A and B are infinite sets. Let N = {1, 2, 3, 4, 5, …, n, …} and W = {0, 1, 2, 3, 4, …, n – 1, n, …}. Clearly, N  W. But it is still possible to establish a one-to-one correspondence between the sets by letting each element in N correspond to a number in W that is one smaller. Thus, N ~ W. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Number of Subsets of a Finite Set 1. If P = {a}, then P has two subsets, Ø and {a}. 2. If Q = {a, b}, then Q has four subsets, Ø, {a}, {b}, and {a, b}. 3. If R = {a, b, c}, then R has eight subsets, Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c} and {a, b, c}. In general, if there are n elements in a set, then 2n subsets can be formed. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.