MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set - PowerPoint PPT Presentation

Counting with Venn Diagrams Story Problems Conclusion MATH 105: Finite Mathematics 6-2: The Number of Elements in a Set Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

Counting with Venn Diagrams Story Problems Conclusion Outline Counting with Venn Diagrams 1 Story Problems 2 Conclusion 3

Counting with Venn Diagrams Story Problems Conclusion Outline Counting with Venn Diagrams 1 Story Problems 2 Conclusion 3

Counting with Venn Diagrams Story Problems Conclusion 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 . Example Find the number of elements in each set.

Counting with Venn Diagrams Story Problems Conclusion 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 . Example Find the number of elements in each set.

Counting with Venn Diagrams Story Problems Conclusion 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 . Example Find the number of elements in each set. (a) A = { 2 , 3 , 5 , a }

Counting with Venn Diagrams Story Problems Conclusion 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 . Example Find the number of elements in each set. (a) A = { 2 , 3 , 5 , a } c ( A ) = 4

Counting with Venn Diagrams Story Problems Conclusion 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 . Example Find the number of elements in each set. (a) A = { 2 , 3 , 5 , a } c ( A ) = 4 (b) B = { 3 , x , y }

Counting with Venn Diagrams Story Problems Conclusion 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 . Example 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

Counting with Venn Diagrams Story Problems Conclusion 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 . Example 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

Counting with Venn Diagrams Story Problems Conclusion 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 . Example 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

Counting with Venn Diagrams Story Problems Conclusion 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 . Example 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

Counting with Venn Diagrams Story Problems Conclusion 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 . Example 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

Counting with Venn Diagrams Story Problems Conclusion 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 ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 5 3 Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 5 3 a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 5 3 a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 x 2 5 3 a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 x 2 5 3 y a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 x 2 5 3 y a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7

Counting with Venn Diagrams Story Problems Conclusion 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 x 2 5 3 y a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7 c ( A ∪ B ) = 6

Counting with Venn Diagrams Story Problems Conclusion 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 x 2 5 3 y a Notice the Relationship. . . c ( A ) + c ( B ) = 4 + 3 = 7 c ( A ∪ B ) = 6 c ( A ∩ B ) = 1

Counting with Venn Diagrams Story Problems Conclusion Counting Rules Counting Formula c ( A ∪ B ) = c ( A ) + c ( B ) − c ( A ∩ B ) Example 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 ).

Counting with Venn Diagrams Story Problems Conclusion Counting Rules Counting Formula c ( A ∪ B ) = c ( A ) + c ( B ) − c ( A ∩ B ) Example 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 ).

Counting with Venn Diagrams Story Problems Conclusion Counting Rules Counting Formula c ( A ∪ B ) = c ( A ) + c ( B ) − c ( A ∩ B ) Example 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 ).

Counting with Venn Diagrams Story Problems Conclusion Counting Rules Counting Formula c ( A ∪ B ) = c ( A ) + c ( B ) − c ( A ∩ B ) Example 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 ).

Counting with Venn Diagrams Story Problems Conclusion Counting Rules Counting Formula c ( A ∪ B ) = c ( A ) + c ( B ) − c ( A ∩ B ) Example 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 ).

Counting with Venn Diagrams Story Problems Conclusion Outline Counting with Venn Diagrams 1 Story Problems 2 Conclusion 3

Counting with Venn Diagrams Story Problems Conclusion Ethnic Foods Example 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 – 50-12 = 38 G ∩ I – 20+32-38 = 14