9 2 Application of Venn Diagrams Venn Diagrams A rectangle is used to represent the universal set Sets within the universal set are depicted usually by circles The region outside a circle but within the rectangle is the complement of that set The combination of any set and its complement gives the universal set The symbol n A denotes the number of elements in A Union Rule for Counting n A B n A n B n A B 1 Let U all the letters of the alphabet and V the vowels draw a Venn Diagram to represent U V and V V b c d f g h j k a e i o u l m n p q r s t v w x y z DeMorgan s Laws A B A B The complement of the union is the intersection of the complements A B A B The complement of the intersection is the union of the complements Given U a b c d e A b c and B b c e find 2 A B 3 A B 4 A B 5 A B To do set operations involving three sets begin inside the parentheses Let U 1 2 3 4 5 6 7 8 9 A 1 2 3 4 5 B 1 2 3 6 8 and C 2 3 4 6 7 6 Find A B C 7 Find A B A C 8 A B C In a Gallop poll 2000 U S adults were asked to agree or disagree with the following statement Job opportunities for women are not equal to those for men The results show that 1190 people agreed with the statement of whom 700 were women Half the people surveyed were women 9 Step 1 Define sets draw diagram W women A agree 10 Step 2 Determine cardinality of each region n U n W n A n W A 11 Step 3 Answer questions from diagram How many men agreed with the statement How many men disagreed FIU MGF1106 9 2 1 of 2 Using Venn Diagrams three intersecting sets separate the universal set into eight regions The numbering of the regions is arbitrary that is any region can be designated as region I 12 Which regions represent set B 13 Which regions represent B C 14 Which regions represent A B 15 Which regions represent B 16 Which regions represent A B C 17 Find A B C 18 Find A B A C 19 Which regions represent B A C 20 Which regions represent A C B C 21 A survey of 1000 people was taken to determine their preferences for Coke Sprite or other carbonated beverages The results were as follows Coke 488 Sprite 464 other carbonated beverage 417 Coke and Sprite 128 Coke and Other 112 Sprite and Other 106 all three 98 Draw a Venn diagram and determine how many people fall into each of the eight regions 22 The following is a description of students at a large university by gender and grade level Gender Female e Male f Total Freshman a 70 100 170 Sophomore b 160 120 280 Junior c 90 170 260 Senior d 80 210 290 Total 400 600 1000 Find a n B F FIU MGF1106 9 2 b n C E c n A D F 2 of 2