2-4 Venn Diagrams & Deductive Reasoning 1. Venn diagrams : Diagram that shows relationships between...
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Transcript of 2-4 Venn Diagrams & Deductive Reasoning 1. Venn diagrams : Diagram that shows relationships between...
2-4
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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