Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets
Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Venn Diagram with Three Sets Verification of Equality of Sets 2.4-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Three Sets: Eight Regions When three sets overlap, it creates eight regions
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region V by finding the elements that are common to all three sets, A ∩ B ∩ C
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region II. Find the elements in A ∩ B and place the elements that are not listed in region V in region II
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region IV. Find the elements in A ∩ C and place the elements that are not listed in region V in region IV
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region VI. Find the elements in B ∩ C and place the elements that are not listed in region V in region VI
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region I by determining the elements in set A that are not in regions II, IV, and V
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region III by determining the elements in set B that are not in regions II, V, and VI
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region VII by determining the elements in set C that are not in regions IV, V, and VI
Copyright 2013, 2010, 2007, Pearson, Education, Inc. General Procedure for Constructing Venn Diagrams with Three Sets, A, B, and C Determine the elements to be placed in region VIII by finding the elements in the universal set that are not in regions I through VII
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} A = { 1, 3, 5, 7, 9} B = {0, 3, 6, 9} C = {0, 2, 3, 7}
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information Step 1: Draw three circles and lable A, B, and C AB C U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information AB C Step 2: Find and place in region V 3 U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 2: Find and place in region II 3 = 9 9 U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 3: Find and place in region IV 3 = 7 9 U 7
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 4: Find and place in region VI 3 = 0 9 U 7 0
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 5: Find and place in region VI 3 = 0 9 U 7 0
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 6: Place remaining A values in region I 3 = 1, 5 9 U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 7: Place remaining B values in region III 3 = 6 9 U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 8: Place remaining C values in region VII 3 = 2 9 U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ex. 1: Create a Venn Diagram using the following information A B C Step 9: Place remaining values in region VIII 3 = 4, 8 9 U
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Verification of Equality of Sets To verify set statements are equal for any two sets selected, we use deductive reasoning with Venn Diagrams. If both statements represent the same regions of the Venn Diagram, then the statements are true for all sets A and B
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Equality of Sets Determine whether (A ⋃ B) ´ = A ´ ⋂ B ´ for all sets A and B
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Equality of Sets Solution Draw a Venn diagram with two sets A and B. Label the regions as indicated
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Equality of Sets Solution
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Equality of Sets Solution Both statements are represented by the same region, IV. Thus (A ⋃ B) ´ = A ´ ⋂ B ´ for all sets A and B
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Use Venn diagrams to determine whether the following statements are equal for all A and B
Copyright 2013, 2010, 2007, Pearson, Education, Inc. De Morgan’s Laws A pair of related theorems known as De Morgan’s laws make it possible to change statements and formulas into more convenient forms. (A ⋂ B)´ = A´ ⋃ B´(A ⋂ B)´ = A´ ⋃ B´ (A ⋃ B)´ = A´ ⋂ B´(A ⋃ B)´ = A´ ⋂ B´
Copyright 2013, 2010, 2007, Pearson, Education, Inc. Homework p. 73 # 5 – 75 (X 5)