dm slide 3.pdf
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Discrete Mathematics Structures
7 1391
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Predicates - the meaning of multiplequantifiers
x
y P(x,y) xy P(x,y)
x
y P(x,y) xy P(x,y)
P(x,y) true for all x, y pairs.
For every value of x we can find a (possibly different)
y so that P(x,y) is true.
P(x,y) true for at least one x, y pair.
There is at least one x for which P(x,y)
is always true.
quantification order is notcommutative.
Suppose P(x,y) = xs favorite class is y.
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Predicates - the meaning of multiplequantifiers
N(x,y) = x is sitting by y
xy N(x,y)
xy N(x,y)
xy N(x,y)
xy N(x,y)
False
True?
True
False
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Proofs - how do you know?
The following statements are true:If I am Mila, then I am a great swimmer.I am Mila.
What do we know to be true?I am a great swimmer!
How do we know it?
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Proofs - how do you know?
A theoremis a statement that can be shown to
be true.Aproofis the means of doing so.
Axiom, postulates,hypotheses and
previously proventheorems.
Rules of inference
Proof
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Proofs - how do you know?
The following statements are true:If I am Mila, then I am a great swimmer.I am Mila.
What do we know to be true?I am a great swimmer!
What rule of
inference can we use
to justify it?
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Proofs - Modus Ponens
I am Mila.
If I am Mila, then I am a great swimmer.
I am a great swimmer!
p
p q
q
Tautology:
(p (p q)) q
InferenceRule:
Modus Ponens
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Proofs - Modus Tollens
I am not a great skater.
If I am Erik, then I am a great skater.
I am not Erik!
q
p q
p
Tautology:
(q (p q)) p
InferenceRule:
Modus Tollens
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Proofs - Addition
I am not a great skater.
I am not a great skater or I am tall.
p
p q
Tautology:
p
(p
q)
InferenceRule:
Addition
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Proofs - Simplification
I am not a great skater andyou are sleepy.
you are sleepy.
p q
p
Tautology:
(p
q)
p
Inference Rule:
Simplification
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Proofs - Disjunctive Syllogism
I am a great eater or I am a great skater.
I am not a great skater.
I am a great eater!
p q
q
p
Tautology:
((p
q)
q)
p
InferenceRule:
DisjunctiveSyllogism
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Proofs - Hypothetical Syllogism
If you are an athlete, you are always hungry.
If you are always hungry, you have a snickers in yourbackpack.
If you are an athlete, you have a snickers in your
backpack.
p q
q
r
p r
Tautology:
((p
q)
(q
r))
(p
r)
Inference Rule:
Hypothetical
Syllogism
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Proofs - A little quiz
Amy is a computer science major.
Amy is a math major or a computer science major.
If Ernie is a math major then Ernie is tall.
Ernie is not tall!
Ernie is not a math major.
Addition
Modus Tollens
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Proofs - A little proof
Heres what you know:
Ellen is a math major or a CS major.If Ellen does not like discrete math, she is not a CSmajor.
If Ellen likes discrete math, she is smart.
Ellen is not a math major.Can you conclude Ellen is smart?
M
CD CD SM
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Proofs - A little proof
1. M C Given
2. D C Given3. D S Given4. M Given
5. C DS (1,4)6. D MT (2,5)
7. S MP (3,6)
Ellen is smart!
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Proofs - A little proof
1. M C Given
2. D C Given3. D S Given4. M Given
5. C Disjunctive Syllogism (1,4)6. C D Contrapositive of 2
7. C S
8. S Modus Ponens (5,7)
Ellen is smart!
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Proof Techniques - direct proofs
A totally different example:
Prove that if n = 3 mod 4, then n2 = 1 mod 4.
7 = 3 mod 4
37 = 1 mod 4
94 = 2 mod 4
16 = 0 mod 4
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Proof Techniques - direct proofs
A totally different example:
Prove that if n = 3 mod 4, then n2 = 1 mod 4.
If n = 3 mod 4, then n = 4k + 3 for some int k.
But then, = (4k + 3)(4k + 3)n2
= 16k2 + 24k + 9
= 16k2 + 24k + 8 + 1
= 4(4k2 + 6k + 2) + 1
= 4j + 1 for some int j
= 1 mod 4.
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Proofs - Fallacies
Rules of inference, appropriately applied givevalidarguments.
Mistakes in applying rules of inference arecalled fallacies.
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Proofs - valid arg or fallacy?
If I am Bonnie Blair, then I skate fast
I skate fast!
I am Bonnie Blair Im Eric Heiden
If you dont give me $10, I bite your ear.
I bite your ear!
You didnt give me $10.
Affirming the
conclusion.
((p q) q) p
Not a tautology.
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