discrete mathematics structures slide 1

7/25/2019 Discrete Mathematics Structures Slide 1

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Discrete Mathematics Structures

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) ) (30(%

)50(%

)20(%

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Expectations

This is really a fun course!

This class contains some of the most beautiful mathyoull ever learn.

Its even useful, beyond giving you techniques to use

solving the puzzles in Games Magazine.

Hints for success Read the textbook.

Lectures really do help! Do the homework.

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Propositional Logic

Whats a proposition?

+ !

+

Apropositionis a declarative statement thats either TRUE or

FALSE (but not both).

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Propositional Logic - negation

Supposepis a proposition.

The negationofpis written pand has meaning:It is not the case thatp.

Ex. DM is NOT Bryans favorite class.

Truth table for negation:

p p

T

F

F

TNotice that

p is aproposition!

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Propositional Logic - conjunction

Conjunction corresponds to English and.

p qis true exactly whenpand qare both true.

Ex. Amy is curious AND clever.

Truth table for conjunction:

p q p

q

T

T

F

F

T

F

T

F

T

F

F

F

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Propositional Logic - disjunction

Disjunction corresponds to English or.

p qis whenpor q(or both) are true.

Ex. Michael is brave OR nuts.

Truth table for disjunction:

p q p q

T

T

F

F

T

F

T

F

T

T

T

F

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Propositional Logic - implication

Implication:p qcorresponds to English ifpthen q, or p

implies q. If it is raining then it is cloudy. If there are 200 people in the room, then I am the Easter

Bunny. Ifpthen 2+2=4.

Truth table for implication:

p q p q

T

T

F

F

T

F

T

F

T

F

T

T

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Propositional Logic - logical equivalence

How many different logical connectives could we define?

To answer, we need the notion oflogical equivalence.

16

How many different logical connectives do we need???

pis logically equivalentto qif their truth tables are the same. We

writep q.

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Propositional Logic - logical equivalence

Challenge: Try to find a proposition that is equivalent top q, but that

uses only the connectives

,

, and

.

p q p q

T

T

FF

T

F

TF

T

F

TT

p q p q p

T

T

FF

T

F

TF

F

F

TT

T

F

TT

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Propositional Logic - proof of 1 famous

Distributivity: p (q r) (p q) (p r)

p q r q r p (q r) p q p r (p q) (p r)

T T T T T T T T

T T F F T T T T

T F T F T T T T

T F F F T T T T

F T T T T T T T

F T F F F T F F

F F T F F F T F

F F F F F F F F

All truthassignment

s forp, q,

and r.

I could say

prove a law of

distributivity.

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Propositional Logic - special definitions

Contrapositives: p q and q p

Ex. If it is noon, then I am hungry.If I am not hungry, then it is not noon.

Converses: p q and qp

Ex. If it is noon, then I am hungry.

If I am hungry, then it is noon.

Inverses: p q and p q

Ex. If it is noon, then I am hungry.

If it is not noon, then I am not hungry.

One of thesethings is not

like the others.

Hint: In oneinstance, the pairof propositions is

equivalent.

p q q p

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Propositional Logic - 2 more defn

A tautologyis a proposition thats always TRUE.

A contradictionis a proposition thats always FALSE.

p p p p p p

T F

F T

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Propositional Logic - say a bit

This session were using propositional logic as a foundation for

formal proofs.

Propositional logic is also the key to writing good codeyoucant do any kind of conditional (if) statement without

understanding the condition youre testing.

All the logical connectives weve discussed are also found inhardware and are called gates.

Well talk about more applications.

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Propositional Logic - an infamous

(p q) q p q

if NOT (blue AND NOT red) OR red then

(p q) q

(p q) q

(p q) q

p (q q)

p q

DeMorgans

Double negation

Associativity

Idempotent

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Propositional Logic - one last proof

( )

( )

substitution for

[p (p q)] q

[(p p) (p q)] q

[p (p q)] q

[ F (p q)] q

(p q) q

(p q) q

(p q) q p (q q )

p T

T

distributive

uniqueness

identity

substitution for

DeMorgansassociative

excluded middle

domination

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Predicate Logic - everybody loves somebody

Proposition, YES or NO?

3 + 2 = 5

X+ 2 = 5

X+ 2 = 5 for any choice of Xin {1, 2, 3}X+ 2 = 5 for some Xin {1, 2, 3}

YES

NO

YES

YES

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Predicate Logic - everybody loves somebody

Alicia eats pizza at least once a week.Garrett eats pizza at least once a week.Allison eats pizza at least once a week.Gregg eats pizza at least once a week.Ryan eats pizza at least once a week.Meera eats pizza at least once a week.

Ariel eats pizza at least once a week.

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Predicates

Alicia eats pizza at least once a week.

Define:EP(x)= xeats pizza at least once a week.Universe of Discourse - x is a student in DM

Apredicate, or propositional function, is afunction that takes some variable(s) asarguments and returns True or False.

Note that EP(x)is not a proposition, EP(Ariel)is.

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Predicates

Suppose Q(x,y)= x> y

Proposition, YES or NO?

Q(x,y)Q(3,4)Q(x,9)

NO

YES

NO

Predicate, YES or NO?

Q(x,y)

Q(3,4)

Q(x,9)

YES

NO

YES

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Predicates - the universal quantifier

Another way of changing a predicate into a proposition.

Suppose P(x) is a predicate on some universe of discourse.Ex. B(x) = x is carrying a backpack, x is set of DM students.

The universal quantifier of P(x) is the proposition:

P(x) is true for all x in the universe of discourse.

We write it x P(x), and say for all x, P(x)

x P(x) is TRUE if P(x) is true for every single x.x P(x) is FALSE if there is an x for which P(x) is false.

x B(x)?

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Predicates - the existential quantifier

Another way of changing a predicate into a proposition.

Suppose P(x) is a predicate on some universe of discourse.Ex. C(x) = x has a candy bar, x is set of DM students.

The existential quantifier of P(x) is the proposition:

P(x) is true for some x in the universe of discourse.

We write it x P(x), and say for some x, P(x)

x P(x) is TRUE if there is an x for which P(x) is true.x P(x) is FALSE if P(x) is false for every single x.

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Predicates - the existential quantifier

B(x) = x is wearing sneakers.

L(x) = x is at least 21 years old.Y(x)= x is less than 24 years old.

Are either of these propositions true?

a) x B(x)b) x (Y(x) L(x))

A: only a is true

B: only b is true

C: both are true

D: neither is true

Universe of discourseis people in this room.

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Predicates - more examples

Universe of discourseis all creatures.

L(x) = x is a lion.

F(x) = x is fierce.C(x) = x drinks coffee.

All lions are fierce.Some lions dont drink coffee.

Some fierce creatures dont drink coffee.

x (L(x) F(x))

x (L(x) C(x))

x (F(x) C(x))

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Predicates - more examples

Universe of discourseis all creatures.

B(x) = x is a hummingbird.

L(x) = x is a large bird.H(x) = x lives on honey.R(x) = x is richly colored.

All hummingbirds are richly colored.

No large birds live on honey.

Birds that do not live on honey are dully colored.

x (B(x) R(x))

x (L(x) H(x))

x (H(x) R(x))

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Predicates - quantifier negation

Not all large birds live on honey.

x P(x) means P(x) is true for every x.What about x P(x) ?

Not [P(x) is true for every x.]

There is an x for which P(x) is not true.x P(x)

So, x P(x) is the same as x P(x).

x (L(x) H(x))

x (L(x) H(x))

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x P(x) means P(x) is true for some x.What about x P(x) ?

Not [P(x) is true for some x.]

P(x) is not true for all x.x P(x)

So, x P(x) is the same as x P(x).

x (L(x) H(x))

x (L(x) H(x))

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So, x P(x) is the same as x P(x).So, x P(x) is the same as x P(x).

General rule: to negate a quantifier, movenegation to the right, changing quantifiers asyou go.

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x (L(x) H(x)) x (L(x) H(x)) Negationrule

x (L(x) H(x)) DeMorgans

x (L(x) H(x)) Subst for

Whats wrong with thisproof?

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discrete mathematics structures slide 1

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