Chapter 1 Introduction to Sets and Logic Mc

Chapter 1 Introduction to Sets and Logic © Mc. Graw-Hill Education. All rights reserved. Authorized only for instructor use in the classroom. No reproduction or further distribution permitted without the prior written consent of Mc. Graw-Hill Education.

Chapter Outline 1. 1 1. 2 1. 3 1. 4 Introduction to Set Theory Subsets and Set Operations Statements and Quantifiers Truth Tables © Mc. Graw-Hill Education.

Section 1. 1 Introduction to Set Theory © Mc. Graw-Hill Education.

Concepts 1. Define Set 2. Write Sets Three Different Ways 3. Define the Empty Set 4. Find the Cardinality of a Set 5. Classify Sets as Finite or Infinite 6. Decide if Two Sets are Equal or Equivalent © Mc. Graw-Hill Education.

Concept 1 Basic Definitions © Mc. Graw-Hill Education.

Sets A set is a collection of objects. In our study of sets, we’ll want to restrict our attention to sets that are well-defined. A set is well -defined if for any given object, we can objectively decide whether it is or is not in the set. Each object in a set is called an element or a member of the set. © Mc. Graw-Hill Education.

Concept 2 Write Sets Three Different Ways © Mc. Graw-Hill Education.

Roster Method The elements of the set are listed between braces, with commas between the elements. The order in which we list elements isn’t important. © Mc. Graw-Hill Education.

EXAMPLE 1 Writing a Set Using the Roster Method (1 of 2) Write the set of months of the year that begin with the letter M. Is this set well-defined? Why or why not? © Mc. Graw-Hill Education.

EXAMPLE 1 Writing a Set Using the Roster Method (2 of 2) SOLUTION The months that begin with M are March and May. So, the answer can be written in set notation as M = {March, May} Each element in the set is listed within braces and is separated by a comma. This is a well-defined set, because every month either begins with M or it does not: there’s no opinions involved. However, if you really wanted to nitpick, you could claim that this set is not well-defined because it didn’t specify a language for the names of months. © Mc. Graw-Hill Education.

Natural Numbers Sets are often labeled with capital letters. The Set of Natural Numbers (Counting Numbers) is defined as N = {1, 2, 3, 4 …}. (When we are designating sets, the three dots, or ellipsis, mean that the list of elements continues indefinitely in the same pattern. ) The Set of Even Natural Numbers is E = {2, 4, 6, 8, …} The Set of Odd Natural Numbers is O = {1, 3, 5, 7, …} © Mc. Graw-Hill Education.

EXAMPLE 2 Writing Sets Using the Roster Method (1 of 2) Use the roster method to do the following: (a) Write the set of natural numbers less than 6. (b) Write the set of odd natural numbers greater than 4. (c) Can you think of another way to describe each set in words? © Mc. Graw-Hill Education.

EXAMPLE 2 Writing Sets Using the Roster Method (2 of 2) SOLUTION (a) {1, 2, 3, 4, 5} (b) {5, 7, 9, 11, . . . } (c) The first set could be described as the set of natural numbers less than or equal to 5, or between 0 and 6. The second set could be described as the set of odd natural numbers greater than 3, or greater than or equal to 5. © Mc. Graw-Hill Education.

Set Notation The symbol is used to show that an object is a member or element of a set. For example, if A is the set of days of the week, we could write Monday A, and read this as “Monday is an element of set A. ” Likewise, we could write Friday A. When an object is not a member of a set, we use the symbol . Since “Icecreamday” is not a day of the week, we can write Icecreamday A, and read this as “Icecreamday is not an element of A. ” © Mc. Graw-Hill Education.

EXAMPLE 3 Understanding Set Notation (1 of 2) Decide whether each statement is true or false. (a) Oregon A, where A is the set of states west of the Mississippi River. (b) 27 {1, 5, 9, 13, 17, . . . } (c) z {v, w, x, y, z} © Mc. Graw-Hill Education.

EXAMPLE 3 Understanding Set Notation (2 of 2) SOLUTION (a) Oregon is west of the Mississippi, so Oregon is an element of A. The statement is true. (b) The pattern shows that each element is 4 more than the previous element. So the next three elements are 21, 25, and 29; this shows that 27 is not in the set. The statement is false. (c) The letter z is an element of the set, so the statement is false. © Mc. Graw-Hill Education.

Three Common Ways to Designate a Set 1. List or Roster Method 2. Descriptive Method 3. Set-Builder Notation © Mc. Graw-Hill Education.

EXAMPLE 4 Describing a Set Using the Descriptive Method Use the descriptive method to describe the set B containing 2, 4, 6, 8, 10, and 12 in two different ways. SOLUTION All of the elements in the set are even natural numbers, and all are less than 14, so B is the set of even natural numbers less than 14. There are plenty of other ways the set could be described. Another is the set of natural numbers between 1 and 15 that are divisible by 2. © Mc. Graw-Hill Education.

Variable The third (and fanciest) method of designating a set is set-builder notation, and this method uses variables. A variable is a symbol (usually a letter) that can represent different elements of a set. © Mc. Graw-Hill Education.

Set-Builder Notation Set-builder notation uses a variable, braces, and a vertical bar | that is read as “such that. ” For example, the set {1, 2, 3, 4, 5, 6} can be written in set-builder notation as { x | x ∈ N and x < 7 } This is read as “the set of elements x such that x is a natural number and x is less than 7. ” We can use any letter or symbol for the variable, but it’s common to use x. © Mc. Graw-Hill Education.

EXAMPLE 5 Writing a Set Using Set-Builder Notation (1 of 2) Use set-builder notation to designate each set, then write how your answer would be read aloud. (a) The set R contains the elements 2, 4, and 6. (b) The set W contains the elements red, yellow, and blue. © Mc. Graw-Hill Education.

EXAMPLE 5 Writing a Set Using Set-Builder Notation (2 of 2) SOLUTION (a) R = {x │ x E and x 7} The set of all x such that x is an even natural number and x is less than 7. (b) W = {x │ x is a primary color} The set of all x such that x is a primary color. © Mc. Graw-Hill Education.

EXAMPLE 6 Using Different Set Notations (1 of 2) Designate the set S with elements 32, 33, 34, 35, … using (a) The roster method. (b) The descriptive method. (c) Set-builder notation. © Mc. Graw-Hill Education.

EXAMPLE 6 Using Different Set Notations (2 of 2) SOLUTION (a) {32, 33, 34, 35, . . . } (b) The set S is the set of natural numbers greater than 31. (c) {x │ x N and x 31} © Mc. Graw-Hill Education.

EXAMPLE 7 Writing a Set Using an Ellipsis Using the roster method, write the set containing all even natural numbers between 99 and 201. SOLUTION {100, 102, 104, . . . , 198, 200} If a set contains many elements, we can again use an ellipsis to represent the missing elements. © Mc. Graw-Hill Education.

Concept 3 Define the Empty Set © Mc. Graw-Hill Education.

Empty Set or Null Set A set with no elements is called an empty set or null set. The symbols used to represent the null set are { } or . © Mc. Graw-Hill Education.

EXAMPLE 8 Identifying Empty Sets (1 of 2) Which of the following sets are empty? (a) The set of woolly mammoth fossils in museums. (b) {x | x is a living woolly mammoth} (c) { } (d) {x | x is a natural number between 1 and 2} © Mc. Graw-Hill Education.

EXAMPLE 8 Identifying Empty Sets (2 of 2) SOLUTION (a) There is certainly at least one woolly mammoth fossil in a museum somewhere, so the set is not empty. (b) Woolly mammoths have been extinct for almost 8, 000 years, so this set is most definitely empty. (c) This one’s tricky. Each of { } and Ø represent the empty set, but {Ø} is the set containing the empty set, which has one element. It’s a goofy set, mind you, but it does have one element, namely the empty set. (d) This set is empty because there are no natural numbers between 1 and 2. © Mc. Graw-Hill Education.

Concept 4 Define the Cardinality of a Set © Mc. Graw-Hill Education.

Cardinal Number The cardinal number of a set is the number of elements in the set. For a set A the symbol for the cardinality is n(A), which is read as “n of A. ” For example, the set R = {2, 4, 6, 8, 10} has a cardinal number of 5 since it has 5 elements. This could also be stated by saying the cardinality of set R is 5. © Mc. Graw-Hill Education.

EXAMPLE 9 Finding the Cardinality of a Set (1 of 2) Find the cardinal number of each set. (a) A = {5, 10, 15, 20, 25, 30} (b) B = {x │ x N and x 16} (c) C = {16} (d) © Mc. Graw-Hill Education.

EXAMPLE 9 Finding the Cardinality of a Set (2 of 2) SOLUTION (a) n(A) = 6 since set A has 6 elements (b) B is the set {1, 2, 3, 4, …, 14, 15}, which has 15 elements. So n(B) = 15. (c) n(C) = 1 since set C has 1 element (d) n( ) = 0 since there are no elements in an empty set © Mc. Graw-Hill Education.

Concept 5 Classify Sets as Finite or Infinite © Mc. Graw-Hill Education.

Finite and Infinite Sets A set is called finite if it has no elements, or has cardinality that is a natural number. A set that is not finite is called an infinite set. The set {p, q, r, s} is a finite set since it has four members: p, q, r, and s. The set {10, 20, 30, . . . } is an infinite set since it has an unlimited number of elements: the natural numbers that are multiples of 10. © Mc. Graw-Hill Education.

EXAMPLE 10 Classifying Sets as Finite or Infinite (1 of 2) Classify each set as finite or infinite. (a) {x │ x N and x 100} (b) Set R is the set of letters used to make Roman numerals. (c) {100, 102, 104, 106, . . . } (d) Set M is the set of people in your immediate family. (e) Set S is the set of songs that can be written. © Mc. Graw-Hill Education.

EXAMPLE 10 Classifying Sets as Finite or Infinite (2 of 2) SOLUTION (a) The set is finite since there are 99 natural numbers that are less than 100. (b) The set is finite since the letters used are C, D, I, L, M, V, and X. (c) The set is infinite since it consists of an unlimited number of elements. (d) The set is finite since there is a specific number of people in your immediate family. (e) The set is infinite because an unlimited number of songs can be written. © Mc. Graw-Hill Education.

Concept 6 Decide if Two Sets are Equal or Equivalent © Mc. Graw-Hill Education.

Equal and Equivalent Sets Two sets A and B are equal (written A = B) if they have exactly the same members or elements. Two finite sets A and B are said to be equivalent (written A B) if they have the same number of elements: that is, n(A) = n(B). © Mc. Graw-Hill Education.

EXAMPLE 11 Deciding If Sets Are Equal or Equivalent (1 of 2) State whether each pair of sets is equal, equivalent, or neither. (a) {p, q, r, s}; {a, b, c, d} (b) {8, 10, 12}; {12, 8, 10} (c) {213}; {2, 1, 3} (d) {1, 2, 10, 20}; {2, 1, 20, 11} (e) {even natural numbers less than 10}; {2, 4, 6, 8} © Mc. Graw-Hill Education.

EXAMPLE 11 Deciding If Sets Are Equal or Equivalent (2 of 2) SOLUTION (a) Equivalent (b) Equal and equivalent (c) Neither (d) Equivalent (e) Equal and equivalent © Mc. Graw-Hill Education.

One-to-One Correspondence Two sets have a one-to-one correspondence of elements if each element in the first set can be paired with exactly one element of the second set and each element of the second set can be paired with exactly one element of the first set. © Mc. Graw-Hill Education.

EXAMPLE 12 Putting Sets in One-to-One Correspondence (1 of 3) Show that … (a) the sets {8, 16, 24, 32} and {s, t, u, v} have a one-to-one correspondence and (b) the sets {x, y, z} and {5, 10} do not have a oneto-one correspondence. Then draw a conclusion about what one-to-one correspondence has to do with equivalence of sets. © Mc. Graw-Hill Education.

EXAMPLE 12 Putting Sets in One-to-One Correspondence (2 of 3) SOLUTION (a) We need to demonstrate that each element of one set can be paired with one and only one element of the second set. One possible way to show a one-to-one correspondence is this: {8, 16, 24, 32} { s, t, u, v } (b) The elements of the sets {x, y, z} and {5, 10} can’t be put in one-to-one correspondence. No matter how we try, there will be an element in the first set that doesn’t correspond to any element in the second set. © Mc. Graw-Hill Education.

EXAMPLE 12 Putting Sets in One-to-One Correspondence (3 of 3) SOLUTION continued What can we conclude? The sets that could be put into one -to-one correspondence had the same number of elements, while the two that could not had a different number of elements. Conclusion? Two sets are equivalent exactly when they can be put into one-to-one correspondence. © Mc. Graw-Hill Education.

Correspondence and Equivalent Sets Two sets are • Equivalent if you can put their elements in one-to -one correspondence. • Not equivalent if you cannot put their elements in one-to-one correspondence. © Mc. Graw-Hill Education.