Modified by Lisa Palen

:
• Diagram that shows relationships between different sets of data.
• can represent conditional statements.
• Every point IN the circle belongs to that set.
• Every point OUT of the circle does not.

All Americans love hot dogs.
Americans People who love hotdogs
Some Martians are green.
Martians Green Aliens
No Martians are Americans.
Martians Americans

• All A’s are B’s. Some A’s are B’s. No A’s are B’s.
• Some B’s are A’s.
A’s B’s
A’s B’s
B’s
A’s
If A, then B.

Let’s see how this works!
Suppose you are given ...
n n

More Venn Practice Problems
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?

http://regentsprep.org/Regents/Math/venn/PracVenn.htm

Law of the Contrapositive
Law of the Contrapositive
A conditional statement and its contrapositive are either both true or both false.
Remember
The contrapositive of p
 q is ~ q

~ p.

Law of the Contrapositive
You are given: Example
If an angle measures 45 º, then it is acute. (a true statement)
You can conclude:
If an angle is not acute, then it does not measure 45 º.

Law of the Contrapositive
Example You are given:
~ t
 a
You can conclude:
~ a
 t

Law of Detachment
You are given: a true conditional statement and the hypothesis occurs
You can conclude: that the conclusion will also occur

Law of Detachment
You are given: Example
If a dog eats biscuits, then he is happy.
Fido eats biscuits.
You can conclude:
Fido is happy.

Law of Detachment
You are given: Example
If a dog eats biscuits, then he is happy.
Fido is happy.
You can conclude:
No conclusion.

Law of Detachment
You are given: Example
If a dog eats biscuits, then he is happy.
Fido is not happy.
Remember the contrapositive:
If a dog is not happy, then he doesn’t eat biscuits.
You can conclude:
Fido does not eat biscuits.

Law of Detachment
You are given: Example
All humans are mortal.
Socrates is a human.
You can conclude:
Therefore, Socrates is mortal.

Law of Detachment
You are given:
All humans are mortal.
Socrates is mortal.
You can conclude:
No conclusion.
Example
(Socrates could be a dog or any other mortal being.)

• You are given: Example
• Those who choose Tint-and-Trim
Hair Salon have impeccable taste.
• You have impeccable taste.
• Can you conclude anything?

Law of
Detachment
You are given: p
 q is true p is given
You can conclude:
Symbolic form q is true

Law of
Syllogism
You are given:
Two true conditional statements and the conclusion of the first is the hypothesis of the second.
You can conclude: that if the hypothesis of the first occurs, then the conclusion of the second will also occur

Law of
Syllogism
You are given:
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.
What is repeated
?
You can conclude:
If it rains today, then we will not see our friends.

Recall
• You are given:
• a = b
• b = c
• What is the conclusion?
• a = c
The name of this algebra property is the
T RANSITIVE PROPERTY

Law of Syllogism
• You are given:
• p
 q
• q
 r
• What is the conclusion?
• First, make a chain.
p  q  r
• You can conclude: p  r

Law of
Syllogism
You are given:
Example
If the dog chases the cat, then the cat will run.
If the cat runs, then the mouse will laugh.
You can conclude:
If the dog chases the cat, then the mouse will laugh.

Law of Syllogism
Example:
If you give a mouse a cookie , then he’s going to ask for a glass of milk .
If you give him the milk , then he’ll probably ask you for a straw .
You can conclude:
If you give a mouse a cookie , then he’ll probably ask you for a straw .

Law of Syllogism
You are given: p
 q
~ t
 a t

~ s
~ s

~ q
What is the conclusion?
First find each contrapositive.
Example
~ q

~ p
~ a
 t s

~ t q
 s p  a
Next, make a chain.
p  q  s  ~ t
 a
You can conclude: