Announcements

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Announcements

• Grading policy
• No Quiz next week
• Midterm next week (Th. May 13).
• The correct answer to the quiz

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Now you can debunk these ?(a sample from the web)
Humans are mammals.Dogs are mammals. Humans are
dogs.Then how am I typing this?
people make mistakesmistakes are wrongalex is a
personalex is wrong
1. Theists (maybe not Jews) define God as
all-powerful. 2. Therefore, God can lift any
size rock. 3. Therefore, there can be no rock
too big for God to lift.4. Therefore, God cannot
create a rock too big for him to lift.5.
Therefore, there is something God cannot do.6.
Therefore, God is not all powerful.7. Therefore
God does not exist (per the definition given by
its believers).
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Lecture 10

• 1.5 Methods of Proof

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1.5 Some Fallacious Proofs
Whats wrong with this?
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1.5
Premise 1 If Portland is the capital of Maine,
then it is in Maine. Premise 2 Portland is in
Maine. Conclusion Portland is the capital of
Maine.
Application of the death penalty is killing a
human being. Killing a human being is wrong.
Therefore, application of the death penalty is
wrong.
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1.5
In this class we will learn the art of proving
theorems. Some names 1) Theorem, Proposition,
Claim, Fact, Result statement that can be
proved. 2) axioms, postulates the basic
assumptions on which the proof us based. 3)
lemma intermediate result to be proved on your
way to proof a theorem. 4) corollary Result that
is directly follows from a theorem you just
proved. 5) Conjecture A Result you think is
true, but cannot prove. We use rules of
inference to prove theorems. By using them wrong,
we create fallacious proofs.
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1.5 Rules of Inference(modus ponens law of
detachment)
always true its a tautology
Conclusion if the premises p and p?q are both
true, then q can only be true. However, if the
premises do not hold, q can still be true or
false.
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1.5 Rules of Inference
simplification
modus tollens
conjunction
addition
disjunctive syllogism
resolution
hypothetical syllogism
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1.5
Examples it snows today If it snows today we go
skiing Therefore we go skiing
If it rains we do not have a barbeque today If we
dont have a barbeque today, well have one
tomorrow Therefore If it rains today, well have
a BBQ tomorrow.
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1.5 Valid arguments.
All inference rules were of the form premise 1
is true, premise 2 is true, therefore conclusion
is true. In general this looks like
For an argument to be true all the premises must
be true.
Example if ngt1 then n2 gt 1 (True) We cannot
conclude (½)2 gt 1 because the premise is not
true.
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Fallacies (revisited)
If you do every problem in this book then youll
learn discrete math. Joe didnot do every problem
in the book, therefore he didnot learn discrete
math.
p you do all problems in the book. q
you learned discrete math.
correct
wrong
fallacy of denying hypothesis
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Fallacies (revisited)
If you do every problem in this book then youll
learn discrete math. Joe learned discrete
math, therefore he did every problem in this
book....
p you do all problems in the book. q
do learned discrete math.
correct
wrong
fallacy of affirming conclusion
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1.5 Inference for Quantified Statements
universal instantiation
universal generalization
existential generalization
existential instantiation
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1.5
Example Everyone in this math class has takes a
CS course Marla is in this class Therefore Marla
has takes a course in CS D(x) x has takes a
math class C(x) x has takes a CS
class. premises
conclusion
Reasoning
universal instantiation
modus ponens
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Some more examples
Example 10 p.74. All movies produced by John
Sayles are wonderful John S. produced a movie
about coal-miners Therefore there a wonderful
movie about coalminers. s(x) x is a movie by
John Sayles c(x) x is a movie about
coalminers w(x) x is a wonderful movie.
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Examples
If it does not rain or it is not foggy then the
sailing race will go on and the lifesaving
demonstration will go on. If the sailing is
held, then the trophy will be awarded and the
trophy was not awarded imply it rained. r
it rains f it is foggy s sailing race goes
on l lifesaving demonstration goes on. t
trophy awarded.
the want the conclusion r to be on the right
side of the arrow
combine premises as much as you can
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Examples
Example 16. S(x,y) is the statement x is shorter
than y. premise
Therefore
What wrong?
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Strategies for proving theorems
Direct proof of implication p?q Assume p true
and use rules of inference to prove that q is
true. Indirect Proof of implication Assume q
is not true, use rules of inference to prove
that p is not true. (NOT q) ? (NOT p) Proof by
contradiction Assume p is not true and use the
rules of inference to prove a contradiction. (NOT
p) ? False
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Direct/Indirect Proofs
Proof the following theorem If n is an odd
integer, then n2 is an odd integer. assume p (n
is an odd integer). n 2k1 for some integer
k, Then n2 (2k1)2 4k2 4k 1 2 (
2k2 2k) 1 2m1, m integer ?
Proof that if 3n2 is odd, then n odd. Assume
(NOT q) n even. Then n 2k, 3n2
6k 2 2(3k1) 2m. Thus, 3n2 is
even. We have proved (NOT q) ? (NOT p)
which is equivalent to p?q ?
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Contradiction
Prove p At least 4 of any 22 days must fall on
the same day of the week (the
pigeonhole principle !). Assume (NOT p). Then at
most 3 days of any 22 are the same day of the
week. This implies that we could only have chosen
3x721 days, which is a contradiction with the
fact that we had chosen 22 days to begin
with. Thus (NOT p) False ? p True.