1. Lesson 2-2
Logic
Lesson 2-2: Logic

2. Venn diagrams: . .A 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.
A =poodle ... a dog
B= horse ... NOT a dog . B
DOGS .A
...B   dog
Example:
Lesson 2-2: Logic

3. 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

4. To Show Relationships using Venn Diagrams:B
A  B
Blue or Brown (includes Purple) … AB
Lesson 2-2: Logic

5. 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

6. Solution:
Twenty-four members of Mu Alpha Theta went to a Mathematics conference.
Three members were on cross country and football teams… CC
Football 3 1 5
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

7. 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 3 5 1 15
Mu Alpha Theta
Lesson 2-2: Logic

8. 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

9. Law of Detachment - Example
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

10. 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

11. Law of Syllogism - 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