# Venn Diagrams and Logic by dffhrtcv3

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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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