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

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

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.

Typical Venn Diagram problem

Venn Practice Problems

Venn Practice Problems 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?

Venn Challenge

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

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: 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

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

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

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

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

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

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?

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

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:

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