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2.3 Venn Diagrams and Set Operations

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2.3 Venn Diagrams and Set Operations Powered By Docstoc
					2.3 Venn Diagrams and Set
       Operations
A Universal Set is a set that contains all the elements
under discussion. It is represented as U.

Describe a universal set that includes all the elements
in sets A and B.
A={horses, cows, pigs} B={goats, sheep}
A possible universal set is farm animals.

Describe a universal set that includes all the elements
in sets C and D.
C={roses, petunias, lilacs}      D={tulips, daisies}
A possible universal set is flowering plants.
In a Venn diagram, the universal set is represented by a
rectangle.
                                     U
Sets within the universal set
are represented by circles.                    A


                                     A’


 The region outside the circle, but inside the rectangle represents
 the elements that are in the universal set but not in A. The
 elements in the universal set that are not in A are called the
 complement of A also written A’.
                  Relations with two sets
 Two sets with no common               U
 elements are called disjoint sets.        B               A


If one set is a proper subset of the  U            A
other, then the Venn diagram will be:          B


 If the two sets are equal (there is   U
 really only one set), we will have            A       B
 this diagram.

 Sets with some common                 U
                                               B           A
 elements will appear like this.
A survey was taken to determine if there will be adequate student support for a blood
drive. Students were asked two questions.
1 “Would you be willing to donate blood?”
2 “Would you be willing to serve breakfast to the blood donors?”



                                Students willing to serve
                                breakfast and give blood
       Students
       willing to give                                               Students
       blood but not                                                 willing to serve
       serve breakfast                                               breakfast but
                                                                     not give blood




        Students not willing to serve breakfast and not willing to give blood
The intersection of two sets is the set of elements that the
two sets have in common.




                    A       C
                        1
                 2 4    3
                        5    7 9
When you see the intersection symbol, think “and”.
The union of two sets is the set of elements that are in either set
or both sets.




                       A        C
                           1
                    2 4    3
                           5     7 9
When you see the union symbol, think “or”.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4, 5}                    U
                                               A
B = {1, 3, 6, 9}                                       B
                                           2
                                                   1
                                           4           6
                                                   3
                                           5           9
Find: A’  B                                               7
                                                           8
A’ = {6, 7, 8, 9}
A’  B = {6, 7, 8, 9}  {1, 3, 6, 9}

A’  B = {6, 9}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4, 5}               U
                                          A       C
C = {2, 4, 6, 8}                      1
                                              2
                                      3           6
                                              4
                                      5           8
Find: A  C ’                                         7
                                                      9
C ’ = {1, 3, 5, 7, 9}

A  C ’ = {1, 2, 3, 4, 5}  {1, 3, 5, 7, 9}
A  C ’ = {1, 3, 5}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}   U     1 3 4 6 10
A = {5, 7, 9}                             A        B
                                          7 5 9   2 8
B = {2, 5, 8, 9}
  A       C
      1
2 4   3
      5   7 9

                6 8



= {1, 3, 5}

 = {2, 4, 6, 7, 8, 9}

= {1, 3, 5, 6, 7, 8, 9}
  A        C
      1
2 4   3
      5     7 9

                  6 8



= {6, 8}

 = {6, 8}

= {1, 3, 5}
     Serve Breakfast
           25
                                      Serve            Give Blood
                                      Breakfast
                  Neither
     Serve          10                            20
                             10          25                15
  Breakfast & Give Blood
  Give Blood      15
      20


70 students were surveyed about their willingness to help with a
blood drive.
20 students said they were willing to serve breakfast and give blood.
25 students said they would serve breakfast but not give blood.
15 students said they would give blood but not serve breakfast.
10 students said they would not help.
How many students said they would be willing to serve breakfast or
give blood?
n(U) = 25
n(A) = 16                  U
                                    A
n(B) = 13                                   B
                                        5
Find: n(A  B)                 11           8

n(AB) = 5                                        2



                               11 + 5 + 8 = 24
                               16 + 13 − 5 = 24

            n(A  B) = n(A) + n(B) − n(AB)
n(A  B) = n(A) + n(B) − n(AB)


n(U) = 37
n(A) = 21             U
                              A
n(B) = 17                              B
                                  15
Find: n(A  B)            6            2

n(AB) = 15                                14



                          21 + 17 − 15 = 23
                          6 + 15 + 2 = 23

				
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