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1-6 Set Theory Warm Up Lesson Presentation Lesson Quiz 1-6 Set Theory Warm Up Write all classifications that apply to each real number. 5 1. rational, repeating decimal 9 2. 5 irrational rational, terminating decimal, integer, 3. 25 whole, natural 4. –6 rational, terminating decimal, integer 7 rational, terminating decimal 5. 10 1-6 Set Theory Sunshine State Standards MA.912.D.7.1 Perform set operations such as union and intersection, complement, and cross product. Also MA.912.D.7.2, MA.912.A.10.1. 1-6 Set Theory Objectives Perform operations involving sets. Use Venn diagrams to analyze sets. 1-6 Set Theory Vocabulary set element union intersection empty set universe complement subset cross product 1-6 Set Theory A set is a collection of items. An element is an item in a set. You can use set notation to represent a set by listing its elements between brackets. The set F of riddles Flore has solved is F = {1, 2, 5, 6}. The set L of riddles Leon has solved is L = {4, 5, 6}. The union of two sets is a single set of all the elements of the original sets. The notation F L means the union of sets F and L. Union F L = {1, 2, 4, 5, 6} 3 Together, Flore and Leon Set F Set L have solved riddles 1, 2, 1 2 56 4 4, 5, and 6. 1-6 Set Theory The intersection of two sets is a single set that contains only the elements that are common to the original sets. The notation F ∩ L means the intersection of sets F and L. Intersection F ∩ L = {5, 6} 3 Flore and Leon have both set F set L solved riddles 5 and 6. 1 2 56 4 The empty set is the set containing no elements. It is symbolized by or {}. 1-6 Set Theory Writing Math In set notation, the elements of a set can be written in any order, but numerical sets are usually listed from least to greatest without repeating any elements. 1-6 Set Theory Additional Example 1A: Finding the Union and Intersection of Sets Find the union and intersection of each pair of sets. A = {5, 10, 15}; B = {10, 11, 12, 13} Set A Set B 5 15 10 11 12 13 To find the union, list every element that lies in one set or the other. A U B = {5, 10, 11, 12, 13, 15} 1-6 Set Theory Additional Example 1A Continued Find the union and intersection of each pair of sets. A = {5, 10, 15}; B = {10, 11, 12, 13} Set A Set B 5 15 10 11 12 13 To find the intersection, list the elements common to both sides. A ∩ B = {10} 1-6 Set Theory Additional Example 1B: Finding the Union and Intersection Find the union and intersection of each pair of sets. A is the set of whole number factors of 15; B is the set of whole number factors of 25. A = {1, 3, 5, 15} Write each set in set B = {1, 5, 25} notation. A U B = {1, 3, 5, 15, 25} To find the union, list all of the elements in either set. A ∩ B = {1, 5} To find the intersection, list the elements common to both sets. 1-6 Set Theory Check It Out! Example 1a Find the union and intersection of each pair of sets. A = {–2, –1, 0, 1, 2}; B = {–6, –4, –2, 0, 2, 4, 6} A U B = {–6, –4, –2, –1, 0, 1, 2, 4, 6} To find the union, list all of the elements in either set. A ∩ B = {–2, 0, 2} To find the intersection, list the elements common to both sets. 1-6 Set Theory Check It Out! Example 1b Find the union and intersection of each pair of sets. A is the set of whole numbers less than 10; B is the set of whole numbers less than 8. A = {1, 2, 3, 4, 5, 6, 7, 8, 9} Write each set in set B = {1, 2, 3, 4, 5, 6, 7} notation. A U B = {0, 1, 2, 3,4, 5, 6, To find the union, list all of 7, 8, 9} the elements in either set. A ∩ B = {0, 1, 2, 3, 4, 5, 6, 7} To find the intersection, list the elements common to both sets. 1-6 Set Theory The universe, or universal set, for a particular situation is the set that contains all of the elements relating to the situation. The complement of set A in universe U is the set of all elements in U that are not in A. Complement of L In the contest described on Universe U 3 slide 6, the universe U is Set F Set L the set of all six riddles. The 1 2 56 4 complement of set L in universe U is the set of all riddles that Leon has not Complement of L = {1, 2, 3}. solved. Leon has not solved riddles 1, 2, and 3. 1-6 Set Theory Additional Example 2A: Finding the Complement of a Set Find the complement of set A in universe U. U is the set of natural numbers less than 10; A is the set of whole-number factors of 9. A = {1, 3 ,9}; U = {1, 2, 3, 4, 5, 6, 7, 8, 9} Draw a Venn diagram to Universe U 8 show the complement of 2 7 set A in universe U Set A 1 3 9 4 Complement of A = {2, 5 6 4, 5, 6, 7, 8} 1-6 Set Theory Additional Example 2B: Finding the Complement of a Set Find the complement of set A in universe U. U is the set of rational numbers; A is the set of terminating decimals. Complement of A = the set of repeating decimals. 1-6 Set Theory Reading Math Finite sets have finitely many elements, as in Example 2A. Infinite sets have infinitely many elements, as in Example 2B. 1-6 Set Theory Check It Out! Example 2 Find the complement of set A in universe U. U is the set of whole numbers less than 12; A is the set of prime numbers less than 12. {0, 1, 4, 6, 8, 9, 10} 1-6 Set Theory One set may be entirely contained within another set. Set B is a subset of set A if every element of set B is an element of set A. The notation B A means that set B is a subset of set A. 1-6 Set Theory Additional Example 3: Determining Relationships Between Sets A is the set of positive multiples of 3, and B is the set of positive multiples of 9. Determine whether the statement A B is true or false. Use a Venn diagram to support your answer. Draw a Venn diagram to show these sets. Set B Set A multiples multiples of 3 that are False; B A of 9 not multiples of 9 1-6 Set Theory Check It Out! Example 3 A is the set of whole-number factors of 8, and B is the set of whole-number factors of 12. Determine whether the statement A B = B is true or false. Use a Venn diagram to support your answer. False; the element 8 of set A, is not an element Set A Set B of set B. 1 3 8 42 6 12 1-6 Set Theory The cross product (or Cartesian product) of two sets A and B, represented by A B, is a set whose elements are ordered pairs of the form (a, b), where a is an element of A and b is an element of B. You can use a chart to find A B. Suppose A = {1, 2} and B = {40, 50, 60}. Set B 40 50 60 1 (1,40) (1,50) (1,60) Set A 2 (2,40) (2,50) (2,60) A B = {(1, 40), (1, 50), (1, 60), (2, 40), (2, 50), (2, 60)} 1-6 Set Theory Additional Example 4: Application The set C = {S, M, L} represents the sizes of cups (small, medium, and large) sold at a frozen yogurt shop. The set F = {V, B, P} represents the available flavors (vanilla, banana, peach). Find the cross product C F to determine all of the possible combinations of sizes and flavors. Set C C F a chart to find the cross Make = S M L {(S, V), (S, B), (S, P), product. (M, V), (M,B), (M, P), V (S,V) (M,V) (L,V) Each pair represents one Set F (L,V), (L, B), (L, P)}; B (S,B) (M,B) (L,B) combination of flavors and 9 possible combinations (S,P) (M,P) (L,P) sizes. P 1-6 Set Theory Check It Out! Example 4 The set MN = {M, N, MN} represents the blood groups in the MN system. Find ABO × MN to determine all of possible blood groups in the ABO × MN systems. M N MN Make a chart to find A (A, M) (A, N) (A, MN) the cross product. B (B, M) (B, N) (B, MN) Each pair represents one combination of AB (AB,M) (AB,N) (AB,MN) ABO and MN blood groups. O (O, M) (O, N) (O, M) ABO MN = {(A, M), (A, N), (A, MN), (B, M),(B, N), (B, MN), (AB, M), (AB, N), (AB, MN), (O, M), (O, N), (O, MN)}: 12 possible blood groups. 1-6 Set Theory Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 1-6 Set Theory Lesson Quiz: Part I 1. Find the union and intersection of sets A and B. A = {4, 5, 6}; B = {5, 6, 7, 8} A U B = {4, 5, 6, 7, 8}; A ∩ B = {5, 6} 2. Find the complement of set C in universe U. U is the set of whole numbers less than 10; C = {0, 2, 5, 6}. {1, 3, 4, 7, 8, 9} 1-6 Set Theory Lesson Quiz: Part II 3. D is the set of whole-number factors of 8, and E is the set of whole-number factors of 24. Determine whether the statement D E is true or false. Use a Venn diagram to support your answer. true Set E 6 Set D 2 4 3 8 1 12 24 1-6 Set Theory Lesson Quiz: Part III 4. Find the cross product F G. F = {–1, 0, 1}; G = {–2, 0, 2} F G = {(–1, –2), (–1, 0), (–1, –2), (0, –2), (0, 0), (0, 2), (1, –2), (1, 0), (1, 2)} 1-6 Set Theory Lesson Quiz for Student Response Systems 1. A set is defined as: A. a collection of items B. a collection of elements C. a union of items D. a union of elements 1-6 Set Theory Lesson Quiz for Student Response Systems 2. The symbol means: A. intersection B. union C. empty set D. set notation 1-6 Set Theory Lesson Quiz for Student Response Systems 3. The intersection: A. contains common elements B. is the empty set C. contains the union D. contains uncommon elements 1-6 Set Theory Lesson Quiz for Student Response Systems 4. Find the intersection of the two sets. A. A B = {1, 3, 4, 5, 6, 7} B. A B = {2} Set A Set B 8 3 C. A B = {2} 4 2 6 1 12 D. A B = {1, 3, 4, 5, 6, 7} 1-6 Set Theory Lesson Quiz for Student Response Systems 5. Find the compliment of set A in universe U. U = All whole-numbers less than 9 A = All even numbers A. {2, 4, 6, 8} B. {1, 3, 6, 7, 8} C. {1, 3, 5, 7, 9} D. {1, 3, 5, 7}

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