Operations on Sets (1).pptx

Operations on Sets Objectives: Union - Define and describe the union and Intersection intersection of sets and the complement of a set. Complement - Use Venn diagrams to represent sets, subsets, and set operations....

Operations on Sets Objectives: Union - Define and describe the union and Intersection intersection of sets and the complement of a set. Complement - Use Venn diagrams to represent sets, subsets, and set operations. Difference - Solve problems involving sets using Venn diagram. 09/09/2024 Operations on Sets 2  The universal set is a general set that contains all elements under discussion. U John Venn (1843 – 1923) created Venn diagrams to show the visual relationship among sets. Universal set is represented by a rectangle Subsets within the universal set are depicted by circles, or sometimes ovals A’ or other shapes. 09/09/2024 Operations on Sets 3  Example 1: Determining Sets From a Venn Diagram Use the Venn diagram to determine each of the following sets: a. U U = { O , ∆ , $, M, 5 } b. A Subtitle A={O,∆} c. The set of elements in U that are not in A. {$, M, 5 }  Representing Two Sets in a Venn Diagram Disjoint Sets: Two sets that have Equal Sets: If A = B then A B no elements in common. and B A. Proper Subsets: All elements of Sets with Some Common Elements set A are elements of set B. Some means “at least one”. The representing the sets must overlap. 09/09/2024 Operations on Sets 5 Example 2: Determining sets from a Venn Diagram Use the Venn Diagram to determine: a. U Solutions: b. B a. U = { a, b, c, d, e, f, g } c. The set of elements in A but not B b. B = {d, e } d. The set of elements in U that are c. {a, b, c } not in B d. {a, b, c, f, g } e. The set of elements in both A and B. e. {d} 09/09/2024 Operations on Sets 6  Insert photo here The Intersection and Union of Sets The intersection of sets A and B, written A∩B, is the set of elements common to both set A and set B. This definition can be expressed in set-builder notation as follows: A∩B = { x | x  A and x B} The union of sets A and B, written AUB is the set of elements are in A or B or in both sets. This definition can be expressed in set-builder notation as follows: AUB = { x | x  A or xB} For any set A: 1. A∩Ø = Ø 2. AUØ = A 09/09/2024 Operations on Sets 7  Example 3: Finding the Intersection of Two Sets Find each of the following intersections: a. {7, 8, 9, 10, 11} ∩ {6, 8, 10, 12} {8, 10} b. {1, 3, 5, 7, 9} ∩ {2, 4, 6, 8} Ø c. {1, 3, 5, 7, 9} ∩ Ø Ø 09/09/2024 Operations on Sets 8  Example 4: Finding the Union of Two Sets Find each of the following Unions: a. {7, 8, 9, 10, 11} {6, 8, 10, 12} {6, 7, 8, 9, 10,11, 12} b. {1, 3, 5, 7, 9} {2, 4, 6, 8} {1, 2, 3, 4, 5, 6, 7, 8} c. {1, 3, 5, 7, 9} Ø {1, 3, 5, 7, 9} 09/09/2024 Operations on Sets 9  Insert photo here The Complement of a Set The complement of set A, symbolized by A’ is the set of all elements in the universal set that are not in A. This idea can be expressed in set-builder notation as follows: A’ = {x | x U and x A} The shaded region represents the complement of set A. This region lies outside the circle. 09/09/2024 Operations on Sets 10 Example 5 Finding a Set’s Complement 1.Let U = { 1, 2, 3, 4, 5, 6,7, 8, 9} and A = {1, 3, 4, 7 }. Find A’. 2. Solution: Set A’ contains all the elements of set U that are not in set A. Because set A contains the elements 1,3,4,and 7, these elements cannot be members of set A’: A’ = {2, 5, 6, 8, 9} 09/09/2024 Operations on Sets 11 Example 6: Performing Set Operations Always perform any operations inside parenthesis first! Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } 1. Find a.(A U B)’ Solution: A U B = {1, 3, 7, 8, 9, 10} (A U B)’ = {2, 4, 5, 09/09/2024 Operations on Sets 12 Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } b.A’ ∩ B’ Solution A’ = {2, 4, 5, 6, 8, 10} B’ = {1, 2, 4, 5, 6, 9} A’ ∩ B’ = {2, 4, 5, 6} 09/09/2024 Operations on Sets 13  Example 7 Determining Sets from a Venn Diagram Set to Description of Set Regions in Venn Determine Diagram a. A B set of elements in A or B or Both I,II,III b. (A B)’ set of elements in U that are not in A B IV c. A B set of elements in both A and B II d. (A B)’ set of elements in U that are not in A B I, III, IV e. A’ B set of elements that are not in A and are in B III f. A B’ set of elements that are in A or not in B or both I,II, IV 09/09/2024 Operations on Sets 14  The Difference of Two Sets The difference of set B from set A, symbolized by A-B. is A B the set of all elements in A, A-B but not in B. 09/09/2024 Operations on Sets 15 Example 8: Find the difference between two sets Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } B = { 3, 7, 8, 10 } 1. Find a. A -B b. B -A Solution: Solution: A - B = {1,9} B - A = {8,10} 09/09/2024 Operations on Sets 16 Example 9: Find the difference between two sets Given: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A = { 1, 3, 7, 9 } C = { 1,2,3,4,5} B = {2,4,6,8,10} D={} Find 5. 6. 1. 7. 2. 09/09/2024 Operations on Sets 17 Example 8: The Cardinal Number of the Union of Two Finite Sets 1. Some of the results of the campus blood drive survey indicated that 490 students were willing to donate blood, 340 students were willing to help serve a free breakfast to blood donors, and 120 students were willing to do both. How many students were willing to donate blood or serve breakfast? 09/09/2024 Operations on Sets 18 Example 9: Fifty people were asked about the pets they keep at home. The Venn diagram shows the result. D = {people who have dogs}, F = {people who have fish}, and C = {people who have cats} How many people have a. dogs? b. dogs and fish? c. dogs or cats? d. Fish and cats but not dogs? e. Dogs or fish but not cats? f. All three? g. Neither one of the three? 09/09/2024 Operations on Sets 19 Example 10: A survey asked 100 students whether they play Mobile Legends(ML) or Call Of Duty (COD) mobile. Eighteen students play ML and COD, 41 play ML and 51 play COD. A. Draw a two-set Venn Diagram that displays the result. B. How many students play COD only? C. How may students do not play either of the online games? 09/09/2024 Operations on Sets 20  Sets and Precise Use of Everyday English Set operations and Venn diagrams provide precise ways of organizing, classifying, and describing the vast array of sets and subsets we encounter every day. Or refers to the union of sets And refers to the intersection of sets 09/09/2024 Operations on Sets 21  The way to get started is to quit talking and begin doing. Walt Disney 09/09/2024 Operations on Sets 22 Thank you 09/09/2024 Operations on Sets 23