1. Venn Diagrams & Deductive Reasoning
G.1cd
Venn Diagrams &
Deductive Reasoning
Modified by Lisa Palen

2. Venn diagrams:
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.

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

4. G1: Venn Diagrams
All A’s are B’s. Some A’s are B’s. No A’s are B’s. Some B’s are A’s
B’s
A’s
B’s
A’s
B’s
A’s
If A, then B.

5. Typical Venn Diagram problem

6. Venn Practice Problems

7. Venn Practice Problems
Let’s see how this works! Suppose you are given ... n n

8. 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?

9. Venn Challenge

10. Venn Practice Problems

11. 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.

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

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

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

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

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

17. Law of Detachment Example
You are given: 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.

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

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

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

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

22. Law of Syllogism Two true conditional statements and
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

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

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

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

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

27. 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:

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