discrete mathematics i lectures chapter 3 dr. adam p. anthony spring 2011 some material adapted from...
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DISCRETE MATHEMATICS ILECTURES CHAPTER 3Dr. Adam P. AnthonySpring 2011
Some material adapted from lecture notes provided by Dr. Chungsim Han and Dr. Sam Lomonaco
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This Week
Introduction to First Order Logic (Sections 3.1—3.3) Predicates and Logic Functions Quantifiers Basic Logic Using Quantifiers Implication, negation rules for quantifiers
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Propositional Functions
Propositional function (open sentence): Statement involving one or more
variables,
e.g.: P(x) = x-3 > 5. Let us call this propositional function
P(x), where P is the predicate and x is the variable.
What is the truth value of P(2) ? false
What is the truth value of P(8) ?
What is the truth value of P(9) ?
false
true
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Propositional Functions
Let us consider the propositional function
Q(x, y, z) defined as:
Q(x, y, z) = x + y = z. Here, Q is the predicate and x, y, and z
are the variables.What is the truth value of Q(2, 3, 5) ? true
What is the truth value of Q(0, 1, 2) ?
What is the truth value of Q(9, -9, 0) ?
false
true
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Function Domains
Propositional functions are just like mathematical functions, they must have a domain: Real numbers Integers People
Students Professors Stock Traders?
Domains are used to clarify the purpose of the predicate Let x be the set of all Students. Let FT(x) = x is a full time
student Sometimes domains are extremely important, particularly
with if-then statements
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Universal Quantification
Let P(x) be a propositional function.
Universally quantified sentence: For all x in the universe of discourse P(x) is true.
Using the universal quantifier : x P(x) “for all x P(x)” or “for every x P(x)”
(Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)
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Universal Quantification
Example: S(x): x is a B-W student. G(x): x is a genius.
What does x (S(x) G(x)) mean ?
“If x is a UMBC student, then x is a genius.” OR “All UMBC students are geniuses.”
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Existential Quantification
Existentially quantified sentence: There exists an x in the universe of discourse
for which P(x) is true.
Using the existential quantifier : x P(x) “There is an x such that P(x).” “There is at least one x such that P(x).”
(Note: x P(x) is either true or false, so it is a proposition, not a propositional function.)
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Existential Quantification
Example: P(x): x is a B-W professor. G(x): x is a genius.
What does x (P(x) G(x)) mean ?
“There is an x such that x is a UMBC professor and x is a genius.”
OR “At least one B-W professor is a genius.”
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Quantifiers, Predicates and Domains
A properly defined quantified statement will have predicates and domains clearly specified How do we say there is a value for x that makes (5x =3)
true? Let x be the set of all real numbers R Let P(x) = (5x = 3) x P(x)
Sometimes, this is more trouble than it’s worth to be this clear so we’ll use shorthand: x in real numbers such that 5x = 3 Or, even shorter: x in R, 5x = 3
Finally, if predicates are used (particularly with implication) but no quantifier is given, then assume is used: P(x) → Q(x) ≡∀x P(x) → Q(x)
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Exercise 2.1.1
Re-write each statement using and (sometimes both!) as appropriate: a) There Exists a negative real x such that x2=8
b) For every nonzero real a, there is a real b such that ab = 1
c) All even integers are positive
d) Some integers are prime
e) If n2=4 then n = 2
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Exercise 2.1.2
Determine the truth values of the following statements: a) For all real numbers x, x2 ≥ 0
b) For all real numbers x, x2 > 0
c) There is an integer n such that n2 = 4
d) There is an integer n such that n2 = 3
e) For all integers x, If x = 2 then x2 = 4
f) If x2 = 4 then x = 2
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Truth Values of Quantified Statements
Take the statement: ∀vertebrates a, Bird(a) → Fly(a) Is it True? Disproof by Counter-example
Take the statement: ∃species s, Pig(s) ∧ Fly(s) Is it True? How do we disprove this one? Disproof by exhaustive search
Picking domains carefully here can make search easier
In Reality, ∀ is a generalized version of AND (∧) and ∃ is a generalized version of OR ∨:• To say ∀x P(x) means we are saying P(x) is true for everything in the world at the same time• ∀x P(x) ≡P(x1) ∧ P(x2) ∧… ∧P(xn)•To say that ∃x P(x) means we are saying that P(x) is true for at least one (or more or ALL) thing in the world• ∃x P(x) ≡P(x1) ∨ P(x2) ∨… ∨P(xn)
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Generalized DeMorgan’s
DeMorgan’s law can apply to longer expressions as long as the connective used is the same throughout: ¬(p ∧ q ∧r ∧z) ≡¬p ∨ ¬q ∨¬r ∨¬z Repeatedly apply associative laws to see
how this works So if ∀and ∃ are just short-hand for ∧
and ∨ then what happens if we negate them?
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Negating Quantified Statements ∀x P(x) ≡P(x1) ∧ P(x2) ∧… ∧P(xn) ∃x P(x) ≡P(x1) ∨ P(x2) ∨… ∨P(xn) ¬(∀x P(x)) ≡¬(P(x1) ∧ P(x2) ∧… ∧P(xn))
≡¬P(x1) ∨ ¬P(x2) ∨… ∨¬P(xn)
≡∃x ¬P(x) ¬(∃x P(x)) ≡¬(P(x1) ∨ P(x2) ∨… ∨P(xn))
≡¬P(x1) ∧ ¬P(x2) ∧… ∧¬P(xn)
≡∀x ¬P(x)
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Tying It All Together
Things seem strange now…logic functions…predicates…quantifiers…
Everything we learned before today is still applicable: Theorem 2.1.1 (laws for simplification) Implication elimination/negation Converse/contrapositive/inverse Any other
equivalences/tautologies/contradictions Truth tables can be used, but less frequently
at this point
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Exercise 2.1.3
Write negations for the following statements: a) For all numbers x, x2 > 0
b) There is an integer n such that n2 = 3
c) All even integers are positive
d) Some integers are prime
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Exercise 2.1.3 cont.
a) For any real x, if x ≥ 0, then x2≥x
b) For any integer n, if n2=n, then n =
c) Some dogs go to hell
d) EVERYBODY fails MTH 161!
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Exercise 2.1.4
Rewrite the following using ∀ and ∃, then determine the truth value of each statement (hint: negating the statement can help—HOW?): a) All even integers are positive
b) Some integers are prime
c) There is a positive real x such that x2 ≥ x3
d) For any real x, if x ≥ 1, then x2 ≥ x
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2.1.4 continued
a) For any integer n, if n2 = n, then n = 0
b) For any real x, if x2 = -1, then x = -1
c) If n2 = 4, then n = 2
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Major Pitfalls with Conditionals Remember how we interpret implication
If you can’t prove me wrong, then I’m right For what things in the world is student(x) →
smart(x) true? Smart students Anybody who is not a student (vacuously true case)
When is ∀people x, student(x) → smart(x) true? When is it false?
When is ∃person x, student(x) → smart(x) true? When is it false? If we meant to say, ‘there exists a student who is
smart’ how do we fix this?
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Common Uses of Quantifiers Universal quantifiers are often used with “implies”
to form “rules”:(x) student(x) smart(x) means “All students are smart”
Universal quantification is rarely used to make blanket statements about every individual in the world: (x)student(x)smart(x) means “Everyone in the world is
a student and is smart” Existential quantifiers are usually used with “and”
to specify a list of properties about an individual:(x) student(x) smart(x) means “There is a student who
is smart”
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Using Multiple Variables, Quantifiers
We already saw a multivariable predicate: Q(x, y, z) = x + y = z.
We can quantify this as (for example): ∃real x∃real y∃real z, such that Q(x,y,z)
Read this as: there exist real number values x, y, and z such that the sum of x and y is z
We can also mix-and-match quantifiers, but it’s trickier and in English it can be confusing: ‘There is a person supervising every detail of the
production process’ Work out on the board
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Understanding Mixed Quantifiers Here’s how you could ‘determine’ the
truth of the following: ∀x in D, ∃y in E such that P(x,y)
Have a friend pick anything in D, then you have to find something in E that makes P(x,y) true
If you ever fail, then the statement is false (counterexample).
∃x in D such that ∀y in E, P(x,y) You need to pick a ‘trump card’: Pick one item
from D such that no matter what someone picks out of E, P(x,y) will be true
Your friend should always fail to prove you wrong
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Exercise 2.2.1
Express the following using ∀ and ∃, then evaluate the truth of the expressiona) For any real x, there is a real y such that x
+ y = 0
b) There is a real x such that for any real y, x ≤ y
c) For any real x, there is a real y such that y < x
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Less Mathematical Practice (2.2.2)Every gardener likes the sun.
x gardener(x) likes(x,Sun) You can fool some of the people all of the time.
x t person(x) time(t) can-fool(x,t)You can fool all of the people some of the time.
x t (person(x) time(t) can-fool(x,t))x (person(x) t (time(t) can-fool(x,t))
All purple mushrooms are poisonous.x (mushroom(x) purple(x)) poisonous(x)
No purple mushroom is poisonous.x purple(x) mushroom(x) poisonous(x) x (mushroom(x) purple(x)) poisonous(x)
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Logic to English Translation (2.2.3))a x person(x) male(x) v female(x)
)b x male(x) ^ person(x)
)c x boy(x) male(x) ^ young(x)
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Negating Mixed Quantifiers
Easy: just apply the negation rule we learned earlier for quantifiers, moving the negation in bit-by-bit: ¬(∀x in D, ∃y in E such that P(x,y))≡∃x in D, ¬(E y in E such that P(x,y))≡∃x in D, ∀y in E such that ¬P(x,y)
Works same for ∃x in D such that ∀y in E, P(x,y) Work out on board!
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Exercise 2.2.4
Negate the following until all negation signs are touching a predicate:
a) ∀x ∀y, P(x,y)
b) ∀x∃y, (P(x) ∧ Z(x,y))
c) ∃x∀y, (P(x) →R(y))
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Order Matters (half the time)! If all your quantifiers are the same, you can
put them in any order and the meaning remains: ∀reals x, ∀ reals y, x + y = y + x
≡∀reals y, ∀ reals x, x + y = y + x Similar for ∃
You have to be VERY careful about the order of mixed quantifiers: What is the difference between:
∀people x, ∃a person y such that loves(x,y)∃person x such that ∀people y, loves(x,y)
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Valid Arguments Using Quantifiers Quantifiers help avoid having to name
everything in the domain But what if we reach a point where we
are looking at a particular item? What can we conclude about that item, if
all we have a quantified statements?
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Universal Instantiation
Rule of Universal Instantiation: If some property is true of
EVERYTHING in a domain, then it is true of any PARTICULAR thing in that domain
x in D, P(x) is TRUE for all things in the domain D
Now, observe an item a from the domain D: Can we conclude anything? P(a) has to be true
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Universal Modus Ponens
x in D, P(x) Q(x)P(a) is true for a particular a in DTherefore, Q(a) is true
Universal instantiation makes this work. How?
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Universal Modus Tollens
x in D, P(x) Q(x)Q(a) for some particular a in DTherefore, P(a)
Same Reasoning about Universal Instantiation here, as well!
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Universal Modus Ponens or Universal Modus Tollens?
a) All good cars are expensiveA smarty is not expensiveTherefore, a smarty is not a good car
b) Any sum of two rational numbers is rationalThe numbers a and b are rationalTherefore, a + b is rational
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Fill In The Blanks (Modus Ponens or Modus Tollens)
If n is even, then n = 2k for some integer k(4x + 2) is evenTherefore, _________________
If m is odd, then m = 2k + 1 for some integer kr 2i + 1 for any integer ITherefore, __________________
n is even if and only if n = 2k for some integer k(m + 1)2 = 2l and l is an integerTherefore, __________________
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Other Quantified Arguments
All of the arguments we looked at in CH 2 have a quantified version of one form or another
Universal Transitivity: x P(x) Q(x)
x Q(x) R(x)x P(x) R(x)
Invalid arguments can be quantified as well, so be careful! Don’t forget about Converse, Inverse error
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Diagrams For Analyzing Arguments All good cars are expensive
A smarty is not an expensive carTherefore, a smarty is not a good car
Expensive Cars
Smarty
Expensive Cars
Good
Cars
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Diagram Example 1
All CS Majors are smartPam is not a CS MajorTherefore, Pam is not smart
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Diagram Example 2
If a product of two numbers is 0, then at least one of the numbers is 0.x 0 and y 0Therefore, xy 0
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Diagram Example 3
No college cafeteria food is goodNo good food is wastedTherefore, No college cafeteria food is wasted
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Diagram Example 4
All teachers occasionally make mistakesNo gods ever make mistakesTherefore, No teachers are gods