Lesson 2-2PracticeModified 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.

All Americans love hot dogs.

Some Martians are green.

No Martians are Americans.

AmericansPeople who love hotdogsVenn Practice ProblemsMartiansGreen AliensMartiansAmericans

G1: Venn DiagramsAll As are Bs. Some As are Bs. No As are Bs.Some Bs are As.BsAsAsBsBsAsIf A, then B.

Typical Venn Diagram problem

Venn Practice Problems

Venn Practice ProblemsLets see how this works! Suppose you are given ...nn

More Venn Practice ProblemsTwenty-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 Problemshttp://regentsprep.org/Regents/Math/venn/PracVenn.htm

Law of the ContrapositiveLaw of the ContrapositiveA conditional statement and its contrapositive are either both true or both false.

RememberThe contrapositive of p q is ~ q ~ p.

Law of the ContrapositiveYou are given: If an angle measures 45, then it is acute. (a true statement)

You can conclude:If an angle is not acute, then it does not measure 45.

Example

Law of the ContrapositiveYou are given: ~ t a

You can conclude:~ a t

Example

Law of DetachmentYou are given:a true conditional statement andthe hypothesis occurs

You can conclude:that the conclusion will also occur

Law of DetachmentYou are given: If a dog eats biscuits, then he is happy.Fido eats biscuits.

You can conclude:Fido is happy.

Example

Law of DetachmentYou are given: If a dog eats biscuits, then he is happy.Fido is happy.

You can conclude:No conclusion.

Example

Law of DetachmentYou 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 doesnt eat biscuits.

You can conclude:Fido does not eat biscuits.

Example

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

Example

Law of DetachmentYou 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

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

Can you conclude anything?

Law of DetachmentYou are given:pq is truep is given

You can conclude:q is true

Symbolic form

Law of SyllogismYou are given:Two true conditional statements andthe 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 SyllogismYou 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.ExampleWhat is repeated?

You are given: a = bb = cWhat is the conclusion?a = cThe name of this algebra property is theT RecallRANSITIVE PROPERTY

You are given: p qq rWhat is the conclusion?First, make a chain.

You can conclude: Law of Syllogismp q rp r

Law of SyllogismExampleYou 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.

Law of SyllogismExample:If you give a mouse a cookie, then hes going to ask for a glass of milk.If you give him the milk, then hell probably ask you for a straw.You can conclude:If you give a mouse a cookie, then hell probably ask you for a straw.

Law of SyllogismExample~ q ~ p~ a t s ~ tq sp q s ~ t ap aYou 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:

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