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Section 2.1

Section 2.1

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Section 2.1

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  1. Math in Our World Section 2.1 The Nature of Sets

  2. Learning Objectives • Define set. • Write sets in three different ways. • Classify sets as finite or infinite. • Define the empty set. • Find the cardinality of a set. • Decide if two sets are equal or equivalent.

  3. Sets A set is a well-defined collection of objects. By well-defined we mean that given any object, we can definitely 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.

  4. EXAMPLE 1 Listing the Elements in a Set Write the set of months of the year that begin with the letter M. 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.

  5. Three methods are commonly used to designate a set: • Roster Form • The elements of the set are listed between braces, with commas between the elements. • Description • This uses a short statement to describe the set. • Set-Builder Notation • This method uses a variable, braces, and a vertical bar | that is read as “such that.” • A variable is a symbol (usually a letter) that can represent different elements of a set.

  6. Sets are generally named with a capital letter. • The Set of Natural Numbers (Counting Numbers) is listed: N = {1, 2, 3, 4 …} • The Integers: I = {…-3, -2, -1, 0, 1, 2, 3…} The three dots, or ellipsis, indicates that the pattern continues indefinitely.

  7. EXAMPLE 2 Writing Sets Using the Roster Method Use the roster method to do the following: (a) Write the set of natural numbers less than 6. (b) Write the set of natural numbers greater than 4. SOLUTION (a) {1, 2, 3, 4, 5} (b) {5, 6, 7, 8, . . .}

  8. Set Notation The symbol  is used to show that an object is a member or element of a set. For example, let set A = {2, 3, 5, 7, 11}. Since 2 is a member of set A, it can be written as 2  {2, 3, 5, 7, 11} or 2 A Likewise, 5  {2, 3, 5, 7, 11} or 5 A When an object is not a member of a set, the symbol  is used. Because 4 is not an element of set A, this fact is written as 4  {2, 3, 5, 7, 11} or 4 A

  9. EXAMPLE 3 Understanding Set Notation 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}

  10. EXAMPLE 3 Understanding Set Notation 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.

  11. EXAMPLE 4 Describing a Set in Words Use the descriptive method to describe the set E containing 2, 4, 6, 8, . . . . SOLUTION The elements in the set are called the even natural numbers. The set E is the set of even natural numbers.

  12. Such that is Set S All elements x The Set of Set-Builder Notation S = { x │ condition(s) }

  13. EXAMPLE 5 Writing a Set Using Set-Builder Notation 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.

  14. EXAMPLE 5 Writing a Set Using Set-Builder Notation SOLUTION • 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.

  15. EXAMPLE 6 Using Different Set Notations Designate the set S with elements 32, 33, 34, 35, … using (a) The roster method. (b) The descriptive method. (c) Set-builder notation. 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}

  16. 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.

  17. Finite and Infinite Sets If a set has no elements or a specific natural number of elements, then it is called a finite set. A set that is not a finite set is called an infinite set. The set {p, q, r, s} is called an finite set since it has four members: p, q, r, and s. The set {10, 20, 30, . . .} is called an infinite set since it has an unlimited number of elements: the natural numbers that are multiples of 10.

  18. EXAMPLE 8 Classifying Sets as Finite or Infinite 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.

  19. EXAMPLE 8 Classifying Sets as Finite or Infinite 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.

  20. 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 .

  21. EXAMPLE 9 Identifying Empty Sets 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}

  22. EXAMPLE 9 Identifying Empty Sets 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) Be careful! Each instance of { } and  represents the empty set, but {} is a set with one element: . (d) This set is empty because there are no natural numbers between 1 and 2.

  23. Cardinal Number The cardinal number of a finite 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.

  24. EXAMPLE 10 Finding the Cardinality of a Set Find the cardinal number of each set. (a) A = {5, 10, 15, 20, 25, 30} (b) B = {10, 12, 14, . . . , 28, 30} (c) C = {16} (d) 

  25. EXAMPLE 10 Finding the Cardinality of a Set SOLUTION (a) n(A) = 6 since set A has 6 elements (b) n(B) = 11 since set B has 11 elements (c) n(C) = 1 since set C has 1 element (d) n() = 0 since there are no elements in an empty set

  26. 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).

  27. EXAMPLE 11 Deciding if Sets are Equal or Equivalent 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}

  28. EXAMPLE 11 Deciding if Sets are Equal or Equivalent SOLUTION (a) Equivalent (b) Equal and equivalent (c) Neither (d) Equivalent (e) Equal and equivalent

  29. 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.

  30. EXAMPLE 12 Putting Sets in One-to-One Correspondence 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 one-to-one correspondence.

  31. EXAMPLE 12 Putting Sets in One-to-One Correspondence 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.

  32. Correspondence vs. Equivalence 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.