??????6 : Predicates, Quantifiers

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??????6 Predicates, Quantifiers
Mathematical Induction

• ??????? (Kuang-Chi Chen)
• chichen6_at_mail.tcu.edu.tw

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Predicates Quantifiers
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Predicates

• The statement X is greater than 3 has 2
  parts object (variable X) and predicate (is
  greater than 3).
• Let P(X) denote statement X is greater than 3,
  where P denote the predicate is greater than 3.
  P(X) is also said to be the value of
  propositional function P() at X .
• For example, let P(X) denote X gt 3, what are
  the truth values of P(4) and P(2)?

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Predicates

• Let Q(x, y) denote the statement x y3, What
  are the truth values of the propositions Q(1, 2)
  and Q(3, 0)?
• Let R(x, y, z) denote the statement xy z,
  What are the truth values of the propositions
  R(1, 2, 3) and R(0, 0, 1)?
• A statement of the form P(x1, x2, , xn) is the
  value of the propositional function P at the
  n-tuple (x1, x2, , xn), and P is also called a
  predicate.

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Predicate Logic

• Predicate logic is an extension of propositional
  logic that permits concisely reasoning about
  whole classes of entities.
• Propositional logic (recall) treats simple
  propositions (sentences) as atomic entities.
• In contrast, predicate logic distinguishes the
  subject of a sentence from its predicate.
• - Remember these English grammar terms?

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Applications of Predicate Logic

• It is the formal notation for writing perfectly
  clear, concise, and unambiguous mathematical
  definitions, axioms, and theorems for any branch
  of mathematics.
• - Predicate logic with function symbols, the
  operator, and a few proof-building rules is
  sufficient for defining any conceivable
  mathematical system, and for proving anything
  that can be proved within that system!

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Other Applications

• Predicate logic is the foundation of thefield of
  mathematical logic, which culminated in Gödels
  incompleteness theorem, which revealed the
  ultimate limits of mathematical thought
• Given any finitely describable, consistent
  proof procedure, there will always remain
• some true statements that will never be
• proven by that procedure.
• i.e., we cant discover all mathematical truths,
  unless we sometimes resort to making guesses.

Kurt Gödel1906-1978
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Practical Applications of Predicate Logic

• It is the basis for clearly expressed formal
  specifications for any complex system.
• It is basis for automatic theorem provers and
  many other Artificial Intelligence systems.
• E.g. automatic program verification systems.
• Predicate-logic like statements are supported by
  some of the more sophisticated database query
  engines and container class libraries.
• - these are types of programming tools.

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Subjects and Predicates

• In the sentence The dog is sleeping
• - The phrase the dog denotes the subject
  the object or entity that the sentence is about.
• - The phrase is sleeping denotes the predicate
  a property that is true of the subject.
• In predicate logic, a predicate is modeled as a
  function P() from objects to propositions.
• - P(x) x is sleeping (where x is any object).

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More About Predicates

• Convention Lowercase variables x, y, z...
  denote objects/entities uppercase variables P,
  Q, R denote propositional functions
  (predicates).
• Keep in mind that the result of applying a
  predicate P to an object x is the proposition
  P(x). But the predicate P itself (e.g. P is
  sleeping) is not a proposition (not a complete
  sentence).
• E.g. if P(x) x is a prime number, P(3) is
  the proposition 3 is a prime number.

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Propositional Functions

• Predicate logic generalizes the grammatical
  notion of a predicate to also include
  propositional functions of any number of
  arguments, each of which may take any grammatical
  role that a noun can take.
• E.g. let P(x, y, z) x gave y the grade z,
  then ifx Mike, y Mary, z A, then
• P(x, y, z) Mike gave Mary the grade A.

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Universes of Discourse (U.D.s)

• The power of distinguishing objects from
  predicates is that it lets you state things about
  many objects at once.
• E.g., let P(x)x1 gt x. We can then say,For
  any number x, P(x) is true instead of (01gt0) ?
  (11gt1) ? (21gt2) ? ...
• - The collection of values that a variable x can
  take is called xs universe of discourse. (domain)

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Quantifiers

• Quantifiers provide a notation to quantify
  (count) how many objects satisfy a given
  predicate.
• There are two types of quantifiers universal
  quantification and existential quantification.

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Two Types of Quantifiers

• ? is the FOR?LL or universal quantifier.?x
  P(x) means for all x in the u.d., P holds.
• ? is the ?XISTS or existential quantifier.?x
  P(x) means there exists an x in the u.d. (that
  is, 1 or more) such that P(x) is true.

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The Universal Quantifier ?

• E.g., let the u.d. of x be parking spaces at TCU.
• Let P(x) be the predicate x is full.
• Then the universal quantification of P(x) is
• ?x P(x), is the proposition
• All parking spaces at TCU are full., i.e.,
• Every parking space at TCU is full., i.e.,
• For each parking space at TCU, that space is
  full.

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The Existential Quantifier ?

• E.g., let the u.d. of x be parking spaces at TCU.
• Let P(x) be the predicate x is full.
• Then the existential quantification of P(x),
• ?x P(x), is the proposition
• Some parking space at TCU is full., i.e.,
• There is a parking space at TCU that is full.
• At least one parking space at TCU is full.

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Free and Bound Variables

• An expression like P(x) is said to have a free
  variable x (meaning, x is undefined).
• A quantifier (either ? or ?) operates on an
  expression having one or more free variables, and
  binds one or more of those variables, to produce
  an expression having one or more bound variables.

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Example of Binding

• P(x,y) has 2 free variables, x and y.
• ?x P(x, y) has 1 free variable, and one bound
  variable. Which is which? y is free
• P(x), where x3 is another way to bind x.
• An expression with zero free variables is a
  bona-fide (actual) proposition.
• An expression with one or more free variables is
  still only a predicate e.g., Q(y) ?x P(x, y).

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Negations

• Let ?x P(x) be the statement of
• Every student in the class has taken a course
  in calculus, where P(x) is x has taken a
  course in calculus
• What is its negation? ?x ?P(x)
• ? (?x P(x)) ?x ?P(x)
• ? (?x Q(x)) ?x ?Q(x)

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Translating from English into Logical Expressions

• Express Every student in this class has studied
  Calculus using predicates and quantifiers.
• Sol.1 Let C() be the predicate of has
  studied Calculus. Assume universe of disclosure
  of x consists of the students in the class.
• Its ?x C(x).

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Translating from English into Logical Expressions
(contd)

• Sol.2 If the u.d. is all people. Let S() be the
  predicate of is in this class. The solution
  is ?x S(x) ? C(x). Not ?x S(x) ? C(x) ?
• Sol.3 We concern about other subjects. Let Q(x,
  y) be the predicate of student x has studied
  subject y. The solution is ?x (S(x) ? Q(x,
  Calculus)).

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Review Predicate Logic

• Objects x, y, z,
• Predicates P, Q, R, are functions mapping
  objects x to propositions P(x).
• Multi-argument predicates P(x, y).
• Quantifiers ?x P(x) For all xs, P(x).
  ?x P(x) There is an x such that P(x).
• Universes of discourse, bound free variables.

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Induction
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Introduction to Induction

• A powerful, rigorous technique for proving that a
  predicate P(n) is true for every natural number
  n, no matter how large.
• Essentially a domino effect principle.

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The First Principle of Mathematical Induction

• Based on a predicate-logic inference rule
• P(0)?n?0 (P(n) ? P(n1))??n?0 P(n)

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The Domino Effect

• Premise 1 Domino 0 falls.
• Premise 2 For every k?N,if domino k falls,
  then so
• does domino k1.
• Conclusion All ofthe dominoes fall down!
• Note this works even if there are
  infinitely many dominoes!

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Validity of Induction

• Proof that ?n?0 P(n) is a valid consequent
• Given any k?0, the 2nd antecedent ?k?0
  (P(k)?P(k1)) trivially implies that ?k?0
  (kltn)?(P(k)?P(k1)), i.e., that (P(0)?P(1)) ?
  (P(1)?P(2)) ? ? (P(n?1)?P(n)).
• Repeatedly applying the hypothetical syllogism
  rule to adjacent implications in this list n-1
  times then gives us P(0)?P(n) which together
  with P(0) (antecedent 1) and modus pones gives
  us P(n). Thus ?n?0 P(n).

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Outline of An Inductive Proof

• Let us say we want to prove ?n P(n)
• Do the base case (or basis step) Prove P(0).
• Do the inductive step Prove ?n P(n)?P(n1).
• E.g. you could use a direct proof, as follows
• Let n?N, assume P(n). (inductive hypothesis)
• Now, under this assumption, prove P(n1).
• The inductive inference rule then gives us
• ?n P(n).

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Generalizing Induction

• Rule can also be used to prove ?n?c P(n) for a
  given constant c?Z, where maybe c?0.
• In this circumstance, the base case is to prove
  P(c) rather than P(0), and the inductive step is
  to prove ?n?c (P(n)?P(n1)).
• Induction can also be used to prove?n?c P(an)
  for any arbitrary series an.
• Can reduce these to the form already shown.

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The Second Principle of Mathematical Induction

• Characterized by another inference rule
• P(0)
  Strong Induction
• ?n?0 (?0?k?n P(k)) ? P(n1)??n?0 P(n)
• The only difference between this and the 1st
  principle is that
• - the inductive step here makes use of the
  stronger hypothesis that P(k) is true for all
  smaller numbers k lt n1, not just for k n.

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Examples of Induction
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1st Principle Example

• Prove that the sum of the first n odd positive
  integers is n2. That is, prove
• Proof by induction
• Base case Let n1. The sum of the first 1 odd
  positive integer is 1 which equals 12.(contd)

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1st Principle Example (contd)

• Inductive step Prove ?? n1 P(n)?P(n1).
• Let n1, assume P(n), and prove P(n1)
• By inductive hypothesis P(n).

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Another Induction Example

• Prove that ??n gt 0, n lt 2n.
• Let P(n) (n lt 2n)
• Base case P(1) (1lt21) (1lt2) T.
• Inductive step For ngt0, prove P(n)?P(n1).
• Assuming n lt 2n, prove n 1 lt 2n1.
• Note n 1 lt 2n 1 (by inductive hypothesis)
• lt 2n 2n (1 lt 2 220 22n-1
  2n)
• 2n1
• So n 1 lt 2n1, and were done.

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More Induction Examples

1. n3 n is divisible by 3
2. 1 2 22 2n 2n1 1
3. Sums of Geometric Progressions
4. An Inequality for Harmonic Numbers
5. , where
6. DeMorgans law
7. Number of Subsets of a finite set
8. 2n lt n!

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More Induction Examples

• E.g. 9 Any chessboard with 2n2n squares but
  with one removed can be tiled using L-shaped
  pieces (which cover 3 squares at a time).
• E.g. 10 The greedy algorithm (selects talk with
  earliest ending time) schedules the most talks in
  a single lecture halls.

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The Second Principle of Mathematical Induction

• Characterized by another inference rule
• P(0)
  Strong Induction
• ?n?0 (?0?k?n P(k)) ? P(n1)??n?0 P(n)
• The only difference between this and the 1st
  principle is that
• - the inductive step here makes use of the
  stronger hypothesis that P(k) is true for all
  smaller numbers k lt n1, not just for k n.

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The 2nd Principle of Math Induction

• P(n) ????,??n????
• 1. ? P(1), P(2), P(3), , P(q) ??,
• 2. ???????? 1?i?k ????
• (??q?k),P(i) ??,
• 3. ? 1 2 ???,? P(k1) ???,
• ? ?n?0 P(n) ???

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2nd Principle Example

• Show that every ngt1 can be written as a product
  ? pi p1p2ps of some series of s prime
  numbers.
• Let P(n) n has that property
• Base case n 2, let s 1, p1 2.
• Inductive step Let n?2. Assume ??2 k n
  P(k).
• Consider n1. If its prime, let s 1, p1
  n1.Else n1 ab, where 2 a n and 2 b
  n.Then a p1p2pt and b q1q2qu.
• Then we have that n1 p1p2pt q1q2qu, a
  product of s tu primes.

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Another 2nd Principle Example

• Prove that every amount of postage of 12 cents or
  more can be formed using just 4-cent and 5-cent
  stamps. P(n) n can be
• Base case 12 3(4), 13 2(4)1(5), 14
  1(4)2(5), 15 3(5), so ?12?n?15, P(n).
• Inductive step Let n15, assume ?12?k?n P(k).
• Note 12?n?3?n, so P(n?3), so add a 4-cent stamp
  to get postage for n1.

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The Well-Ordering Property

• Another way to prove the validity of the
  inductive inference rule is by the well-ordering
  property, which says that
• - Every non-empty set of non-negative integers
  has a minimum (smallest) element.
• - ? ??S?N ?m?S ?n?S m?n
• This implies that n?P(n) (if non-empty) has a
  minimal element m, but then the assumption that
  P(m-1)?P((m-1)1) would be contradicted.

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The Well-Ordering Property

• Any non-empty subset of Z contains a smallest
  element. (Z is well ordered)
• Example 1 If a is an integer and d is a positive
  integer. There are unique integers q and r with
  0?rltd and a dq r.
• Example 2 If there is a cycle of length m (m?3)
  among the players in a round-robin tournament,
  there must be a cycle of three.

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The Method of Infinite Descent

• A way to prove that P(n) is false for all n?N.
• (Sort of a converse to the principle of
  induction)
• Prove first that ?P(n) ?kltn P(k).
• - Basically, For every P there is a smaller P.
• But by the well-ordering property of N, we know
  that ?P(m) ? ?P(n) ?P(k) nk.
• - Basically, If there is a P, there is a
  smallest P.
• Note that these are contradictory unless ?P(m),
• - that is, ?m?N P(m). There is no P.

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Infinite Descent

• Infinite DescentFor a propositional function
  P(n), P(k) is false for all positive integers.
• Example is irrational.

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Infinite Descent Example

• Theorem 21/2 is irrational.
• Proof Suppose 21/2 is rational, then ?m, n?Z
  21/2m/n. Let M, N be the m, n with the least n.
  
• So ?kltN, j 21/2 j/k (let j 2N-M, k M-N).
  

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? ?Propositional Equivalence
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Propositional Equivalence

• Compound propositions p and q that have same
  truth values in all possible cases are called
  logically equivalent, denoted as p q.
• The symbol is not a logical connective, it
  means that p q is not a compound prop.
• Sometimes, the symbol ? is used instead of .

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Tautologies and Contradictions

• A tautology is a compound proposition that is
  true no matter what the truth values of its
  atomic propositions are!
• Ex. p ? ?p Truth table
• A contradiction is a compound proposition that is
  false no matter what!
• Ex. p ? ?p Truth table
• Other compound props. are contingencies.

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Logical Equivalence

• Compound proposition p is logically equivalent to
  compound proposition q, written p q, IFF the
  compound proposition p?q is a tautology.
• Compound propositions p and q are logically
  equivalent to each other IFF p and q contain the
  same truth values as each other in all rows of
  their truth tables.

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Prove Equivalence via Truth Tables

• Ex. Prove that p?q ?(?p ? ?q).

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Equivalence Laws

• These are similar to the arithmetic identities
  you may have learned in algebra, but for
  propositional equivalences instead.
• They provide a pattern or template that can be
  used to match all or part of a much more
  complicated proposition and to find an
  equivalence for it.

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Example of Equivalence Laws

• Identity p?T p p?F p
• Domination p?T T p?F F
• Idempotent p?p p p?p p
• Double negation ??p p
• Commutative p?q q?p p?q q?p
• Associative (p?q)?r p?(q?r)
  (p?q)?r p?(q?r)

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More Equivalence Laws

• Distributive p?(q?r) (p?q)?(p?r)
  p?(q?r) (p?q)?(p?r)
• De Morgans ?(p?q) ?p ? ?q ?(p?q) ?p ? ?q
  
• Trivial tautology/contradiction p ? ?p T
  p ? ?p F

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Nested Quantifiers
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Nested Quantifiers

• Example Let the u.d. of x y be people.
• Let L(x, y) x likes y (a predicate with 2
  free variables)
• Then ?y L(x, y) There is someone whom x
  likes. (A predicate with 1 free variable, x)
• Then ?x (?y L(x, y)) Everyone has someone
  whom they like.(A proposition with 0 free
  variables.)

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Quantifier Exercise

• If R(x,y)x relies upon y, express the
  following in unambiguous English
• ?x(?y R(x,y))
• ?y(?x R(x,y))
• ?x(?y R(x,y))
• ?y(?x R(x,y))
• ?x(?y R(x,y))

Everyone has someone to rely on.
Theres a poor overburdened soul whom everyone
relies upon (including himself)!
Theres some needy person who relies upon
everybody (including himself).
Everyone has someone who relies upon them.
Everyone relies upon everybody, (including
themselves)!
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Natural Language Is Ambiguous

• Everybody likes somebody.
• For everybody, there is somebody they like,
• ?x ?y Likes(x, y)
• or, there is somebody (a popular person) whom
  everyone likes?
• ?y ?x Likes(x, y)
• Somebody likes everybody.
• Same problem Depends on context, emphasis.

Probably more likely
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More Conventions

• Sometimes the universe of discourse is restricted
  within the quantification,
• E.g., ?xgt0 P(x) is shorthand for
• For all x that are greater than zero, P(x).
• ?x (xgt0 ? P(x))
• ?xgt0 P(x) is shorthand for
• There is an x greater than zero such that
  P(x).
• ?x (xgt0 ? P(x))

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More About Binding

• ?x ?x P(x) - x is not a free variable in ?x
  P(x), therefore the ?x binding isnt used.
• (?x P(x)) ? Q(x) - The variable x is outside of
  the scope of the ?x quantifier, and is therefore
  free. Not a proposition!
• (?x P(x)) ? (?x Q(x)) This is legal, because
  there are 2 different xs!

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Quantifier Equivalence Laws

• Definitions of quantifiers If u.d.a,b,c, ?x
  P(x) ? P(a) ? P(b) ? P(c) ? ?x P(x) ? P(a) ?
  P(b) ? P(c) ?
• From those, we can prove the laws?x P(x) ? ??x
  ?P(x)?x P(x) ? ??x ?P(x)
• Which propositional equivalence laws can be used
  to prove this?

DeMorgan's
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(Chalkboard)

• (Chalkboard) Another way to see why the order of
  quantifiers matters is to expand out the
  definitions of FORALL and EXISTS in terms of AND
  and OR. E.g., suppose the universe of discourse
  just consists of two objects a and b. Now,
  consider some predicate P(x, y). Then,
• FORALL x EXISTS y P(x, y)
• ? (EXISTS y P(a, y)) /\ (EXISTS y P(b, y))
• ? (P(a, a) \/ P(a, b)) /\ P(b, a) \/ P(b, b)).
• In contrast,
• EXISTS y FORALL x P(x, y)
• ? (FORALL x P(x, a)) \/ (FORALL x P(x, b))
• ? (P(a, a) /\ P(b, a)) \/ (P(a, b) /\ P(b, b)).

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(Chalkboard)

• To see that these two are inequivalent, suppose
  only P(a, a) and P(b, b) are true. Then, the
  first proposition (with the FORALL first) is
  true, but, the second proposition (with the
  EXISTS first) is true. Students can come up with
  this counterexample in-class as an exercise.

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More Equivalence Laws

• ?x ?y P(x,y) ? ?y ?x P(x,y)?x ?y P(x,y) ? ?y ?x
  P(x,y)
• ?x (P(x) ? Q(x)) ? (?x P(x)) ? (?x Q(x))?x (P(x)
  ? Q(x)) ? (?x P(x)) ? (?x Q(x))
• Exercise See if you can prove these yourself.
• What propositional equivalences did you use?

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More Notational Conventions

• Quantifiers bind as loosely as neededparenthesiz
  e ?x P(x) ? Q(x)
• Consecutive quantifiers of the same type can be
  combined ?x ?y ?z P(x,y,z) ??x,y,z P(x,y,z)
  or even ?xyz P(x,y,z)
• All quantified expressions can be reducedto the
  canonical alternating form ?x1?x2?x3?x4 P(x1,
  x2, x3, x4, )

( )
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Defining New Quantifiers

• As per their name, quantifiers can be used to
  express that a predicate is true of any given
  quantity (number) of objects.
• Define ?!x P(x) to mean P(x) is true of exactly
  one x in the universe of discourse.
• ?!x P(x) ? ?x (P(x) ? ??y (P(y) ? y? x))There
  is an x such that P(x), where there is no y such
  that P(y) and y is other than x.

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Some Number Theory Examples

• Let u.d. the natural numbers 0, 1, 2,
• A number x is even, E(x), if and only if it is
  equal to 2 times some other number.?x (E(x) ?
  (?y x2y))
• A number is prime, P(x), iff its greater than 1
  and it isnt the product of two non-unity
  numbers.?x (P(x) ? (xgt1 ? ??yz xyz ? y?1 ?
  z?1))

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Goldbachs Conjecture (unproven)

• Using E(x) and P(x) from previous slide,
• ?E(xgt2) ?P(p),P(q) pq x
• or, with more explicit notation
• ?x xgt2 ? E(x) ?
• ?p ?q P(p) ? P(q) ? pq x.
• Every even number greater than 2 is the sum
  of two primes.

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Calculus Example

• One way of precisely defining the calculus
  concept of a limit, using quantifiers

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Deduction Example

• Definitions s Socrates (ancient Greek
  philosopher) H(x) x is human M(x) x
  is mortal.
• Premises H(s) Socrates
  is human.
• ?x H(x)?M(x) All humans are mortal.

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Deduction Example Continued

• Some valid conclusions you can draw
• H(s)?M(s) Instantiate universal. If
  Socrates is human
  then he is mortal.
• ?H(s) ? M(s) Socrates is
  inhuman or mortal.
• H(s) ? (?H(s) ? M(s)) Socrates is human,
  and also either inhuman or mortal.
• (H(s) ? ?H(s)) ? (H(s) ? M(s)) Apply
  distributive law.
• F ? (H(s) ? M(s))
  Trivial contradiction.
• H(s) ? M(s)
  Use identity law.
• M(s)
  Socrates is mortal.

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Another Example

• Definitions H(x) x is human M(x) x
  is mortal G(x) x is a god
• Premises
• ?x H(x) ? M(x) (Humans are mortal) and
• ?x G(x) ? ?M(x) (Gods are immortal).
• Show that ??x (H(x) ? G(x)) (No human is a
  god.)

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The Derivation

• ?x H(x)?M(x) and ?x G(x)??M(x).
• ?x ?M(x)??H(x) Contrapositive.
• ?x G(x)??M(x) ? ?M(x)??H(x)
• ?x G(x)??H(x) Transitivity of ?.
• ?x ?G(x) ? ?H(x) Definition of ?.
• ?x ?(G(x) ? H(x)) DeMorgans law.
• ??x G(x) ? H(x) An equivalence law.

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Summary of Predicate Logic

• From these sections you should have learned
• - Predicate logic notation conventions
• - Conversions predicate logic ? clear English
• - Meaning of quantifiers, equivalences
• - Simple reasoning with quantifiers