Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x...
Transcript of Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x...
![Page 1: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/1.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Set operationsand
Venn Diagrams
![Page 2: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/2.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Set operations and Venn diagrams
A ! B = { x | x " A and x " B }
This is the intersection of A and B.
A # B = { x | x " A or x " B }
This is the union of A and B.
An element of A ! B belongs to both A
and B,
an element of A # B is required to
belong to at least one of the sets.
![Page 3: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/3.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A ! B = { x | x " A and x " B }
A B
![Page 4: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/4.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A # B = { x | x " A or x " B }
A B
![Page 5: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/5.jpg)
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A = {1,2,3,4}, B = {1,3,5,7}, and C = {7,9,3}, andthe universal set U = {1,2,3,4,5,6,7,8,9}. Locate allthis information appropriately in a Venn diagram.
Sets and the Universal Set
![Page 6: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/6.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
A = {1,2,3,4}, B = {1,3,5,7}, and C = {7,9,3}, andthe universal set U = {1,2,3,4,5,6,7,8,9}. Locate allthis information appropriately in a Venn diagram.
With each number,place it in theappropriate region.
Sets and the Universal Set
![Page 7: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/7.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
1
A = {1,2,3,4}, B = {1,3,5,7}, and C = {7,9,3}, andthe universal set U = {1,2,3,4,5,6,7,8,9}. Locate allthis information appropriately in a Venn diagram.
With each number,place it in theappropriate region.
Sets and the Universal Set
![Page 8: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/8.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
12
A = {1,2,3,4}, B = {1,3,5,7}, and C = {7,9,3}, andthe universal set U = {1,2,3,4,5,6,7,8,9}. Locate allthis information appropriately in a Venn diagram.
With each number,place it in theappropriate region.
Sets and the Universal Set
![Page 9: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/9.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
123
A = {1,2,3,4}, B = {1,3,5,7}, and C = {7,9,3}, andthe universal set U = {1,2,3,4,5,6,7,8,9}. Locate allthis information appropriately in a Venn diagram.
With each number,place it in theappropriate region.
Sets and the Universal Set
![Page 10: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/10.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A = {1,2,3,4}, B = {1,3,5,7}, and C = {7,9,3}, andthe universal set U = {1,2,3,4,5,6,7,8,9}. Locate allthis information appropriately in a Venn diagram.
A B
C
With each number,place it in theappropriate region.
Check out the Venndiagram and makesure you agree withwhere all theelements have beenplaced.
123
5
6
7
89
4
Sets and the Universal Set
![Page 11: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/11.jpg)
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Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
![Page 12: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/12.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
![Page 13: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/13.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
(A # B) ! (A # C)
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
![Page 14: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/14.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
(A # B) ! (A # C)
A B
C
A # (B ! C)
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
![Page 15: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/15.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
A B
C
(A # B) ! (A # C)
A B
C
A # (B ! C)
![Page 16: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/16.jpg)
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DeMorgan’s Law
(A # B)c = Ac ! Bc
![Page 17: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/17.jpg)
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A B
(A # B)c is the
gray region in thispicture
DeMorgan’s Law
(A # B)c = Ac ! Bc
![Page 18: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/18.jpg)
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A B
(A # B)c is the
gray region in thispicture
A B
At this stage A ispainted green.
DeMorgan’s Law
(A # B)c = Ac ! Bc
![Page 19: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/19.jpg)
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A B
(A # B)c is the
gray region in thispicture
A B
Now B is painted red, sowhat’s outside of A andoutside of B hasn’t beenpainted.
DeMorgan’s Law
(A # B)c = Ac ! Bc
![Page 20: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/20.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
(A # B)c is the
gray region in thispicture
A B
What hasn’t beenpainted is Ac ! Bc
DeMorgan’s Law
(A # B)c = Ac ! Bc
![Page 21: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/21.jpg)
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DeMorgan’s Other Law
(A ! B)c = Ac # Bc
![Page 22: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/22.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
DeMorgan’s Other Law
(A ! B)c = Ac # Bc
A B
![Page 23: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/23.jpg)
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Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(a) students studying math but not business
![Page 24: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/24.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(a) students studying math but not business
M ! Bc
![Page 25: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/25.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(b) students studying both math and business
![Page 26: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/26.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(b) students studying both math and business
M ! B
![Page 27: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/27.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(c) students studying either math or business
![Page 28: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/28.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(c) students studying either math or business
M # B
![Page 29: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/29.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(d) students who study neither math norbusiness
![Page 30: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/30.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(d) students who study neither math norbusiness
(M # B)c or Mc ! Bc
![Page 31: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/31.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(e) students who don’t study math and who don’tstudy business
![Page 32: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/32.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Grouping StudentsLet’s denote by M and B the students in aparticular university that are studyingmathematics and business. Write down the setthat describes each of the following groups ofstudents:
(e) students who don’t study math and who don’tstudy business
(M # B)c or Mc ! Bc
![Page 33: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/33.jpg)
COPYRIGHT © 2006 by LAVON B. PAGE
Sketching regions representing sets:
Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
![Page 34: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/34.jpg)
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Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
Sketching regions representing sets:
![Page 35: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/35.jpg)
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Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
Sketching regions representing sets:
![Page 36: Set operations and Venn Diagrams - University of Hawaii operations and Venn diagrams A ! B = { x | x " A and x " B } This is the intersection of A and B. A # B = { x | x " A or x "](https://reader031.fdocuments.in/reader031/viewer/2022022500/5aa21ccb7f8b9ac67a8caf52/html5/thumbnails/36.jpg)
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Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
Sketching regions representing sets: