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

Deductive vs. Inductive Reasoning The difference: inductive reasoning uses patterns to arrive at a conclusion (conjecture) deductive reasoning uses facts, rules, definitions or properties to arrive at a conclusion.

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.

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

Typical Venn Diagram problem

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

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

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

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.

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

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

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

The Law of Syllogism If p q and q r, then p  r. If the conclusion of the first conditional is the hypothesis of the second conditional, then the first hypothesis implies that the second conclusion will be true.

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?

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

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:

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:

Example 1 If Tim gets stung by a bee, then he will get very ill. If he gets very ill, then he will go to the hospital. Tim gets stung by a bee. Conclusion? Tim will go to the hospital. Law of Syllogism

Example 2 If Hank applies for the job, then he will be the new lifeguard at the pool. If he is the new lifeguard at the pool, then he will buy a new car. Hank applies for the job. Conclusion? Hank will buy a new car. Law of Syllogism

Example 3 If two planes intersect, then their intersection is a line. Plane A and plane B intersect. Conclusion? Plane A and plane B intersect in a line. Law of Detachment

Example 4 If you cut class, then you will receive a detention. You cut class. Conclusion? You received a detention. Law of Detachment

Example 5 If Jay doesn’t work hard, then he won’t start the game and will quit the team. Jay quit the team. Conclusion? No conclusion. We do not why he quit the team.

Re Cap A proof is a convincing logical argument that uses deductive reasoning. Law of Detachment: If a conditional is true and its hypothesis is true, then the conclusion is also true. Law of Syllogism: If p q and q r, then p  r.