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Venn Diagrams &Deductive Reasoning

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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.

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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

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G1: Venn DiagramsAll 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

A’s B’sB’sA’s

If A, then B.

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Typical Venn Diagram problem

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Venn Practice Problems

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Venn Practice Problems

Let’s see how this works!Suppose you are given ...

n n

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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?

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Venn Challenge

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http://regentsprep.org/Regents/Math/venn/PracVenn.htm

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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.

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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

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Law of the Contrapositive

You are given: ~ t a

You can conclude:~ a t

Example

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Law of Detachment

You are given:a true conditional statement andthe hypothesis occurs

You can conclude:that the conclusion will also occur

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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

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Law of Detachment

You are given: If a dog eats biscuits, then he is

happy.Fido is happy.

You can conclude:No conclusion.

Example

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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

doesn’t eat biscuits.

You can conclude:Fido does not eat biscuits.

Example

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Law of Detachment

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

Example

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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

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

You have impeccable taste.

Can you conclude anything?

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Law of Detachment

You are given:pq is truep is given

You can conclude:

q is true

Symbolic form

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Law of Syllogism

You 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

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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 repeate

d?

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Recall

You are given: a = b b = c What is the conclusion? a = cThe name of this algebra property is the

T

RANSITIVE PROPERTY

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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

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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

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Law of SyllogismExample: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.

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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 ~ tq s

p q s ~ t a

p a

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