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Section 2.4 Venn Diagrams with Three Sets and Verification of Equality of Sets. What You Will Learn. Venn Diagram with Three Sets Verification of Equality of Sets. Three Sets: Eight Regions. When three sets overlap, it creates eight regions. - PowerPoint PPT Presentation
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.
2.4-3
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
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
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
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
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
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
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
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
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
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Blood Types
2.4-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionBlood 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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
SolutionI 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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Verification of Equality of SetsTo 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
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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Example 3: Equality of SetsSolutionDraw a Venn diagram with two sets A and B. Label the regions as indicated.
2.4-19
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Example 3: Equality of Sets
Solution
2.4-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Equality of Sets
SolutionBoth statements are represented by the same region, IV.
Thus (A ⋃ B)´ = A´ ⋂ B´ for all sets A and B.
2.4-21
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´
2.4-22