DM Slide 3.pdf

7/25/2019 DM Slide 3.pdf

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Discrete Mathematics Structures

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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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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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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 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 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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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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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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