discrete mathematics structures slide 1
TRANSCRIPT
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Discrete Mathematics Structures
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) ) (30(%
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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.
x C(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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Predicates - quantifier negation
No large birds live on honey.
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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Predicates - quantifier negation
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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Predicates - quantifier negation
No large birds live on honey.
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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