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Lecture 1.2: Equivalences, and Predicate Logic
CS 250, Discrete Structures, Fall 2015
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren, Zeph Grunschlag
p is logically equivalent to q if their truth tables are the
same. We write p q.
In other words, p is logically equivalent to q if p q is True.
There are some famous laws of equivalence, which we review next. Very useful in simplifying complex composite propositions
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Predicate Logic 5
Laws of Logical Equivalences
Identity laws
Like adding 0
Domination laws
Like multiplying by 0
Idempotent laws
Delete redundancies
Double negation
“I don’t like you, not”
Commutativity
Like “x+y = y+x”
Associativity
Like “(x+y)+z = y+(x+z)”
Distributivity
Like “(x+y)z = xz+yz”
De Morgan
Rosen; page 27: Table 6
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Predicate Logic 6
De Morgan’s Law - generalized
De Morgan’s law allow for simplification of negations of complex expressions
Conjunctional negation:
(p1p2…pn) (p1p2…pn)
“It’s not the case that all are true iff one is false.”
Disjunctional negation:
(p1p2…pn) (p1p2…pn)
“It’s not the case that one is true iff all are false.”
Let us do a quick (black board) proof to show that the law holds
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Predicate Logic 7
Another equivalence example
(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
DeMorgan’s
Double negation
Associativity
Idempotent
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Yet another example
Show that [p (p q)] q is a tautology.
We use to show that [p (p q)] q T.
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
De Morgan’s
associative
uniqueness
domination
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Predicate Logic – why do we need it?
Proposition, YES or NO?
3 + 2 = 5
X + 2 = 5
X + 2 <= 5 for any choice of X in {1, 2, 3}
X + 2 = 5 for some X in {1, 2, 3}
Propositional logic can not handle statements such as the last two.
Predicate logic can.
YES
NO YES
YES
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Predicate Logic Example
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 -- Definition
Alicia eats pizza at least once a week. Define: P(x) = “x eats pizza at least once a week.” Universe of Discourse - x is a student in cs250 A predicate, or propositional function, is a function that
takes some variable(s) as arguments and returns True or False.
Another way of changing a predicate into a proposition. Suppose P(x) is a predicate on some universe of discourse. 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. Ex: B(x) = “x is carrying a backpack,” x is a set of cs250 students.
x B(x)?
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The Universal Quantifier - example
B(x) = “x is allowed to drive a car” L(x) = “x is at least 16 years old.” Are either of these propositions true? a) x (L(x) B(x)) b) x B(x)
A: only a is true
B: only b is true
C: both are true
D: neither is true
Universe of discourse is people in this room.
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The Existential Quantifier
Another way of changing a predicate into a proposition. Suppose P(x) is a predicate on some universe of discourse. 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.
Ex. C(x) = “x has black hair,” x is a set of cs 250 students.
x C(x)?
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The Existential Quantifier - example
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 discourse is people in this room.
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Predicate Logic 17
Predicates – more examples
Universe of discourse is all creatures.
L(x) = “x is a lion.” F(x) = “x is fierce.” C(x) = “x drinks coffee.” All lions are fierce.
Some lions don’t drink coffee.
Some fierce creatures don’t drink coffee.
x (L(x) F(x))
x (L(x) C(x))
x (F(x) C(x))
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Predicates – some more example
Universe of discourse is all birds.
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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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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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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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, move negation to the
right, changing quantifiers as you go.
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Some quick questions
p T =? (what law?)
p T =? (what law?)
(p q) = ? (what law?)
P(x) = “x >3” P(x) is a proposition: yes or no?
P(3) is a proposition: yes or no? If yes, what is its value?
P(4) is True or False?
If the universe of discourse is all natural numbers, what is the value of
x P(x)
x P(x)
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Today’s Reading and Next Lecture
Rosen 1.3 and 1.4
Please start solving the exercises at the end of each chapter section. They are fun.
Please read 1.4 and 1.5 in preparation for the next lecture