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Lesson 2-2
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Lesson 2-2: Logic 1
Lesson 2-2
Logic
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Lesson 2-2: Logic 2
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: A =poodle ... a dog
B= horse ... NOT a dog
.BDOGS
.A...B dog
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Lesson 2-2: Logic 3
For all..., every..., if...then...All right angles are congruent.
Congruent Angles
Right Angles
Example1:
Example 2: Every rose is a flower.
Flower
Rose
Example 3: If two lines are parallel, then they do not intersect.
lines that do not intersect
parallel lines
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Lesson 2-2: Logic 4
To Show Relationships using Venn Diagrams:
Blue or Brown (includes Purple) … AB
A B
A B
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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...
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Lesson 2-2: Logic 6
Solution:
Three members were on cross country and football teams…
3The above sentence tells you two draw overlapping circles and put 3 in CCF
Twenty-four members of Mu Alpha Theta went to a Mathematics conference.
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.
CC Football
5
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.
1
Continued….
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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
35 1
Band
15
Mu Alpha Theta
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Lesson 2-2: Logic 8
Law of Detachment
Given:
a true conditional statement and
the hypothesis occurs
pq is true
p is given
Conclusion:
the conclusion will also occur
q is true
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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 1:
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.
Example 2:
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Lesson 2-2: Logic 10
Law of Syllogism
Given:Two true conditional statements andthe conclusion of the first is the hypothesis of the second.
pq and qr
Conclusion:If the hypothesis of the first occurs, then the conclusion of the
second will also occur.pr
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Lesson 2-2: Logic 11
Law of Syllogism - Example
If it rains today, then we will not see our friends.
Example:
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:
Given: