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Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets

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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Venn Diagram with Three Sets Verification of Equality of Sets 2.4-2
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Three Sets: Eight Regions When three sets overlap, it creates eight regions. 2.4-3
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  • 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. 2.4-4
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  • 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. 2.4-5
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  • 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. 2.4-6
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  • 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. 2.4-7
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  • 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. 2.4-8
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  • 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. 2.4-9
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  • 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. 2.4-10
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  • 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. 2.4-11
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Human blood is classified (typed) according to the presence or absence of the specific antigens A, B, and Rh in the red blood cells. Antigens are highly specified proteins and carbohydrates that will trigger the production of antibodies in the blood to fight infection. Blood containing the Rh antigen is labeled positive, +, while blood lacking the Rh antigen is labeled negative, . Example 2: Blood Types 2.4-12
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Blood lacking both A and B antigens is called type O. Sketch a Venn diagram with three sets A, B, and Rh and place each type of blood listed in the proper region. A person has only one type of blood. Example 2: Blood Types 2.4-13
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: Blood Types 2.4-14
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Solution Blood containing Rh is is + Blood not containing Rh is Example 2: Blood Types All blood in the Rh circle is + All blood outside the Rh circle is Intersection of all 3 sets, V, is AB+ II contains only A and B, AB 2.4-15
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Solution I contains A only, A III contains B only, B Example 2: Blood Types IV is A+ VI is B+ VII contains only Rh antigen, O+ VIII lacks all three antigens, O 2.4-16
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  • 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. 2.4-17
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Equality of Sets Determine whether (A B) = A B for all sets A and B. 2.4-18
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  • 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. 2.4-19
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Equality of Sets Solution 2.4-20
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  • 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. 2.4-21
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  • Copyright 2013, 2010, 2007, Pearson, Education, Inc. De Morgans Laws A pair of related theorems known as De Morgans 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 2.4-22