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COPYRIGHT © 2006 by LAVON B. PAGE
Set operationsand
Venn Diagrams
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.
COPYRIGHT © 2006 by LAVON B. PAGE
A ! B = { x | x " A and x " B }
A B
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A # B = { x | x " A or x " B }
A B
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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
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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
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
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
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
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
COPYRIGHT © 2006 by LAVON B. PAGE
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
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A B
C
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
COPYRIGHT © 2006 by LAVON B. PAGE
A B
C
(A # B) ! (A # C)
Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
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)
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Distributive Law for Unions
A # (B ! C) = (A # B) ! (A # C)
A B
C
(A # B) ! (A # C)
A B
C
A # (B ! C)
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DeMorgan’s Law
(A # B)c = Ac ! Bc
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A B
(A # B)c is the
gray region in thispicture
DeMorgan’s Law
(A # B)c = Ac ! Bc
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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
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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
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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
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DeMorgan’s Other Law
(A ! B)c = Ac # Bc
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DeMorgan’s Other Law
(A ! B)c = Ac # Bc
A B
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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
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
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
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
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
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
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
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
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
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
COPYRIGHT © 2006 by LAVON B. PAGE
Sketching regions representing sets:
Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
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Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
Sketching regions representing sets:
COPYRIGHT © 2006 by LAVON B. PAGE
Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
Sketching regions representing sets:
COPYRIGHT © 2006 by LAVON B. PAGE
Sketch the region corresponding to the set
(A # Bc) ! C
A B
C
Sketching regions representing sets:
