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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

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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Page 1: 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.

Section 2.4

Venn Diagrams

with Three Sets and

Verification of Equality

of Sets

Page 2: 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

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Page 3: 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.

Three Sets: Eight Regions

When three sets overlap, it creates eight regions.

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Page 4: 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.

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.

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Page 5: 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.

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.

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Page 6: 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.

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.

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Page 7: 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.

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

Page 8: 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.

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

Page 9: 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.

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.

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Page 10: 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.

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

Page 11: 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.

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.

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Page 12: 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.

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

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Page 13: 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.

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

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Page 14: 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.

Example 2: Blood Types

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Page 15: 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.

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–

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Page 16: 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.

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

Page 17: 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.

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

Page 18: 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.

Example 3: Equality of Sets

Determine whether (A ⋃ B)´ = A´ ⋂ B´ for all sets A and B.

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Page 19: 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.

Example 3: Equality of SetsSolutionDraw a Venn diagram with two sets A and B. Label the regions as indicated.

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Page 20: 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.

Example 3: Equality of Sets

Solution

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Page 21: 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.

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

Page 22: 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.

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´

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