MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set
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1 MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006
2 Outline 1 Counting with Venn Diagrams 2 Story Problems 3 Conclusion
3 Outline 1 Counting with Venn Diagrams 2 Story Problems 3 Conclusion
4 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set.
5 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set.
6 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a}
7 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4
8 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4 (b) B = {3, x, y}
9 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4 (b) B = {3, x, y} c(b) = 3
10 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4 (b) B = {3, x, y} c(b) = 3 (c) A B
11 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4 (b) B = {3, x, y} c(b) = 3 (c) A B c(a B) = 1
12 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4 (b) B = {3, x, y} c(b) = 3 (c) A B c(a B) = 1 (d) A B
13 Counting Set Elements Number of Elements in a Set Let A be a set. Then, c(a) is the number of elements in the set A. Find the number of elements in each set. (a) A = {2, 3, 5, a} c(a) = 4 (b) B = {3, x, y} c(b) = 3 (c) A B c(a B) = 1 (d) A B c(a B) = 6
14 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. Notice the Relationship... c(a) + c(b) = = 7
15 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B Notice the Relationship... c(a) + c(b) = = 7
16 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 Notice the Relationship... c(a) + c(b) = = 7
17 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 3 Notice the Relationship... c(a) + c(b) = = 7
18 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B Notice the Relationship... c(a) + c(b) = = 7
19 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 Notice the Relationship... c(a) + c(b) = = 7
20 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 Notice the Relationship... c(a) + c(b) = = 7
21 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 x Notice the Relationship... c(a) + c(b) = = 7
22 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 x y Notice the Relationship... c(a) + c(b) = = 7
23 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 x y Notice the Relationship... c(a) + c(b) = = 7
24 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 x y Notice the Relationship... c(a) + c(b) = = 7 c(a B) = 6
25 Placing Elements in a Venn Diagram Note how the elements of A = {2, 3, 5, a} and B = {3, x, y} are arranged in a Venn Diagram. A B 2 a 5 3 x y Notice the Relationship... c(a) + c(b) = = 7 c(a B) = 6 c(a B) = 1
26 Counting Rules Counting Formula c(a B) = c(a) + c(b) c(a B) Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(a C). (b) Let N = {0, 1, 2,...} be the set of natural numbers. Find c(n). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(d).
27 Counting Rules Counting Formula c(a B) = c(a) + c(b) c(a B) Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(a C). (b) Let N = {0, 1, 2,...} be the set of natural numbers. Find c(n). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(d).
28 Counting Rules Counting Formula c(a B) = c(a) + c(b) c(a B) Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(a C). (b) Let N = {0, 1, 2,...} be the set of natural numbers. Find c(n). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(d).
29 Counting Rules Counting Formula c(a B) = c(a) + c(b) c(a B) Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(a C). (b) Let N = {0, 1, 2,...} be the set of natural numbers. Find c(n). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(d).
30 Counting Rules Counting Formula c(a B) = c(a) + c(b) c(a B) Each of the next examples leads to another useful counting rule. (a) If A = {2, 3, 5, a} and C = {1, 4, b} find c(a C). (b) Let N = {0, 1, 2,...} be the set of natural numbers. Find c(n). (c) Suppose that U = {1, 2, 3, 4, 5} and D = {2, 4, 5}. Find c(d).
31 Outline 1 Counting with Venn Diagrams 2 Story Problems 3 Conclusion
32 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 12 G I = 38 G I = 14
33 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food 50 G I 12 G I = 38 G I = 14
34 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 12 G I = 38 G I = 14 G 50
35 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 12 G I = 38 G I = 14 G I 50
36 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 12 G I = 38 G I = 14 G I 50 12
37 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 12 G I = 38 G I = 14 G I 50 12
38 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 50 G I 12 G I = 38 G I =
39 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 50 G I 12 G I = 38 G I =
40 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 50 G I 12 G I = 38 G I =
41 Ethnic Foods Fifty people are interviewed about their food preferences. Twenty of them like Greek food, 32 like Italian food, and 12 like neither Greek nor Italian food. How many like Greek but not Italian food? G Greek food I Italian food G I 50 G I 12 G I = 38 G I =
42 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone R W 37 R W = 4 R W 63-4 = 59 R W = 400
43 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone 500 R W 37 R W = 4 R W 63-4 = 59 R W = 400
44 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone R W 37 R W = 4 R W 63-4 = 59 R W = 400 W 500
45 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone R W 37 R W = 4 R W 63-4 = 59 R W = 400 W 500 R
46 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone R W 37 R W = 4 R W 63-4 = 59 R W = 400 W R
47 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone R W 37 R W = 4 R W 63-4 = 59 R W = 400 W R
48 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone W 500 R R W 37 R W = R W 63-4 = 59 R W = 400
49 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone W 500 R R W 37 R W = R W 63-4 = 59 R W =
50 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone W 500 R R W 37 R W = R W 63-4 = 59 R W =
51 Newspaper Subscriptions A survey of 500 families provided the following data: 63 subscribe to the Wall Street Journal, 41 subscribe to Rolling Stone, and 37 of the families who subscribe to Rolling Stone do not subscribe to the Wall Street Journal. How many families subscribe to both, and how many subscribe to neither? W Wall Street Journal R Rolling Stone W 500 R R W 37 R W = R W 63-4 = 59 R W =
52 Car Sales Of the cars sold during the month of July, 90 had air conditioning, 100 had automatic transmissions, and 75 had power steering. Five cars had all three of these extras. Twenty cars had none of these extras. Twenty cars had only air conditioning; 60 cars had only automatic transmissions; and 30 cars had only power steering. Ten cars had both automatic transmission and power steering. (a) How many cars had both power steering and air conditioning? (b) How many had both automatic transmission and air conditioning? (c) How many cars were sold in July?
53 Student Transportation The transportation and Parking Committee at Gigantic State University collects data from 100 students on how they commute to campus. The following data is obtained: 8 drive a car at least part of the time 20 use the bus at least part of the time 48 ride a bicycle at least part of the time 38 do none of these no student who drives a care also uses the bus How many students who ride a bicycle also dirve a car or use the bus?
54 Outline 1 Counting with Venn Diagrams 2 Story Problems 3 Conclusion
55 Important Concepts Things to Remember from Section Do not double count elements in a union. 2 Counting Formula #1: c(a B) = c(a) + c(b) c(a B) 3 Counting Formula #2: c(u) = c(a) + c(a) 4 Only place numbers on Venn Diagrams if they belong to a single area.
56 Important Concepts Things to Remember from Section Do not double count elements in a union. 2 Counting Formula #1: c(a B) = c(a) + c(b) c(a B) 3 Counting Formula #2: c(u) = c(a) + c(a) 4 Only place numbers on Venn Diagrams if they belong to a single area.
57 Important Concepts Things to Remember from Section Do not double count elements in a union. 2 Counting Formula #1: c(a B) = c(a) + c(b) c(a B) 3 Counting Formula #2: c(u) = c(a) + c(a) 4 Only place numbers on Venn Diagrams if they belong to a single area.
58 Important Concepts Things to Remember from Section Do not double count elements in a union. 2 Counting Formula #1: c(a B) = c(a) + c(b) c(a B) 3 Counting Formula #2: c(u) = c(a) + c(a) 4 Only place numbers on Venn Diagrams if they belong to a single area.
59 Important Concepts Things to Remember from Section Do not double count elements in a union. 2 Counting Formula #1: c(a B) = c(a) + c(b) c(a B) 3 Counting Formula #2: c(u) = c(a) + c(a) 4 Only place numbers on Venn Diagrams if they belong to a single area.
60 Next Time... Venn Diagrams are useful for organizing known information about set sizes, but we don t always know that information. In the next section we look at the first of several counting rules used to determine set sizes. For next time Read Section 6-3 (pp ) Prepare for quiz on 6-1 and 6-2 Do Problem Sets 6-1 A; 6-2 A,B
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