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Venn Diagrams and Logic

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					Lesson 2-2


Logic

   Lesson 2-2: Logic   1
     Venn diagrams:
   show relationships between different sets of data.
   can represent conditional statements.
   is usually drawn as a circle.
    •   Every point IN the circle belongs to that set.
    •   Every point OUT of the circle does not.

Example:       .B                       A =poodle ... a dog
                      DOGS
                                        B= horse ... NOT a dog
                       .A
                                         ...B   dog
                                 Lesson 2-2: Logic            2
    For all..., every..., if...then...
Example1: All right angles are congruent.
                                               Congruent Angles

Example 2: Every rose is a flower.
                                                 Right Angles
 Flower                  lines that do
                         not intersect
  Rose

                         parallel lines



Example 3:   If two lines are parallel, then they do not intersect.

                           Lesson 2-2: Logic              3
To Show Relationships using Venn Diagrams:




       A          B


           A B

     Blue or Brown (includes Purple) … AB


                      Lesson 2-2: Logic   4
  Example:
Twenty-four members of Mu Alpha Theta went to a Mathematics
  conference.
One-third of the members ran cross country.
One sixth of the members were on the football team .
Three members were on cross country and football teams.
The rest of the members were in the band.
How many were in the band?

Hint: Draw a Venn Diagram and take one sentence at a time...


                            Lesson 2-2: Logic          5
                          Twenty-four members of Mu Alpha Theta
     Solution:            went to a Mathematics conference.

   Three members were on cross                          CC        Football
    country and football teams…
                                                         5     3       1
The above sentence tells you two draw
overlapping circles and put 3 in CCF

One-third    of the members ran cross country.
24 / 3 = 8; 8 members run cross country. So put 5 in cross country as there are
already 3 in cross country.
One   sixth of the members were on the football team .
24/6 = 4; 4 members play football. So put 1 in football as there are already 3 in
football.
                                                              Continued….
                                     Lesson 2-2: Logic                  6
     Example: Continued……
   The rest of the members were in the band. How many were in
    the band?
   Out of 24 members in Mu Alpha Theta, 9 play football or
    run cross country.
   Therefore, 15 are in the band.

                                      CC             Football
                  Band
                    15
                                        5        3        1


                    Mu Alpha Theta
                             Lesson 2-2: Logic                  7
Law of Detachment
 Given:
 a true conditional statement and
 the hypothesis occurs
                             pq is true
                             p is given

 Conclusion:
 the conclusion will also occur
                              q is true

                      Lesson 2-2: Logic    8
   Law of Detachment - Example

Example 1:
Given: If three points are collinear, then the points are all on one line.
      E, F, and G are collinear.
Conclusion: E, F, and G are all on one line.


Example 2:

 Given:   If I find $20 in the street, then I’ll take you to the movies.
          On October 10 I found $20 in the street.
 Conclusion: I will take you to the movies.

                                Lesson 2-2: Logic             9
   Law of Syllogism
Given:
Two true conditional statements and
the conclusion of the first is the hypothesis of the second.
                       pq and qr

Conclusion:
If the hypothesis of the first occurs, then the conclusion of the
    second will also occur.
                                pr


                             Lesson 2-2: Logic             10
      Law of Syllogism - Example

  Example:

  Given:      If it rains today, then we will not have a picnic.

              If we do not have a picnic, then we will not see our
              friends.


Conclusion:    If it rains today, then we will not see our friends.



                              Lesson 2-2: Logic            11

				
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