G 1 cd Venn Diagrams Deductive Reasoning 1

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

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

2 Venn diagrams: Diagram that shows relationships between different sets of data. can represent

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. Americans People who love hotdogs

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

4 G 1: Venn Diagrams All A’s are B’s. Some A’s are B’s. No

4 G 1: 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 If A, then B. B’s A’s B’s

5 Typical Venn Diagram problem

5 Typical Venn Diagram problem

6 Venn Practice Problems

6 Venn Practice Problems

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

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

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

9 Venn Challenge

10 Venn Practice Problems http: //regentsprep. org/Regents/Math/venn/Prac. Venn. htm

10 Venn Practice Problems http: //regentsprep. org/Regents/Math/venn/Prac. Venn. htm

11 Law of the Contrapositive A conditional statement and its contrapositive are either both

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: Example If an angle measures 45º,

12 Law of the Contrapositive You are given: Example 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.

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

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

14 Law of Detachment You are given: a true conditional statement and the hypothesis

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

15 Law of Detachment You are given: Example If a dog eats biscuits, then

15 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.

16 Law of Detachment You are given: Example If a dog eats biscuits, then

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

17 Law of Detachment Example You are given: If a dog eats biscuits, then

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 You are given: All humans are mortal. Socrates is a

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

19 Law of Detachment You are given: All humans are mortal. Socrates is mortal.

19 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. )

20 Example You are given: Those who choose Tint-and-Trim Hair Salon have impeccable taste.

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 You are given: p q is true p is given

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

22 Law of Syllogism You are given: Two true conditional statements and the conclusion

22 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 23 You are given: Example If it rains today, then we

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

Recall 24 You are given: a = b b = c What is the

Recall 24 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 You are given: p q q r What is the

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

Law of Syllogism 26 You are given: Example If the dog chases the cat,

Law of Syllogism 26 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.

27 Law of Syllogism Example: If you give a mouse a cookie, then he’s

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: If you give a mouse a cookie, then he’ll probably ask you for a straw.

28 Law of Syllogism Example You are given: ~ q ~ p p q

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