2-4 Venn Diagrams & Deductive Reasoning 1
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. 2
All Americans love hot dogs. Some Martians are green. No Martians are Americans. Americans People who love hotdogs Venn Practice Problems Martians Green Aliens Martians Americans 3
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 If A, then B. 4
Typical Venn Diagram problem 5
Venn Practice Problems 6
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Let’s see how this works! Suppose you are given... n n 8
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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? 10
Venn Challenge 11
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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. 14
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 15
Law of the Contrapositive You are given: ~ t a You can conclude: ~ a t Example 16
Law of Detachment You are given: a true conditional statement and the hypothesis occurs You can conclude: that the conclusion will also occur 17
Law of Detachment You are given: If a dog eats biscuits, then he is happy. Fido eats biscuits. You can conclude: Fido is happy. Example 18
Law of Detachment You are given: If a dog eats biscuits, then he is happy. Fido is happy. You can conclude: No conclusion. Example 19
Law of Detachment 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. Example 20
Law of Detachment You are given: All humans are mortal. Socrates is a human. You can conclude: Therefore, Socrates is mortal. Example 21
Law of Detachment 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 22
Example You are given: Those who choose Tint-and-Trim Hair Salon have impeccable taste. You have impeccable taste. Can you conclude anything? 23
Law of Detachment You are given: p q is true p is given You can conclude: q is true Symbolic form 24
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 25
Law of Syllogism You are given: 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. Example What is repeated ? 26
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 27
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 28
Law of Syllogism You are given: 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. Example 29
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. 30
Law of Syllogism 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: Example ~ q ~ p ~ a t s ~ t q s p q s ~ t a p a 31