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Chapter 1. The Foundations

Chapter 1. The Foundations. Cheng-Chia Chen. 1. The foundations. Logic: The basis of all mathematical reasoning has practical applications to the design of computing machine give the precise meaning of math. statements. Sets: The basis of all mathematical structures.

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Chapter 1. The Foundations

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  1. Chapter 1. The Foundations Cheng-Chia Chen

  2. 1. The foundations • Logic: • The basis of all mathematical reasoning • has practical applications to the design of computing machine • give the precise meaning of math. statements. • Sets: • The basis of all mathematical structures. • definitions, operations • Functions : • definitions, classification, operations

  3. 1.1 Logic • Proposition: • A statement that is either true or false (,but not both). • Example: 1. Taipei is the capital of ROC. 2. Tokyo is the capital of Korea. 3. 1+1 = 2. 4. 2+2 = 3. 5. There is no integer x,y,z and n > 2 s.t. xn + yn = zn ---- Fermat’s last theorem 6. “Every even number is the sum of two prime numbers.” ---- Goldaach’s conjecture (1742) ==> 1. Statement (1)~(5) are propositions, 2. (1),(3) are true; (2),(4) are false. 3. Nobody knew (5) is true or false until 1994, but it must be true or false, hence it is a proposition. (6) open problem.

  4. More Examples • Examples: 1. What time is it? (interrogative) 2. Read this carefully! (command) 3. x+1 = 2. 4. x+y = z. ==> 1. (1) and (2) are not statements 2. (3) and (4) are not propositions: neither true nor false!

  5. Propositional variables and propositional constants • In algebra, we use variables to denote an item with unspecified value. • e.g. 10 chickens and rabbits in a cage with 16 feet. ==> #rabbits = ? and #chickens=? • ==> let x = #rabbits, y = #chickens • ==> x + y = 10 /\ 2x + 4y = 16. • Likewise, In logic, we can use propositional variable to denote an arbitrary proposition with unspecified truth value. • e.g., If it is raining, the ground is wet. since the ground is wet, it is raining now. • ==> Let p =def “It is raining”, q =def “the ground is wet” • ==> ((p ->q) /\ q) -> p. • Propositional constants: only two values: • T(true), F(false).

  6. Forming Compound propositions from simpler ones • Primitive propositions: • propositional variables • propositional constants • P, Q: two known propositions • Then the followings are also propositions: 1. “P and Q” : ( P /\ Q) conjunction of P and Q 2. “It is not the case that p”: (~P) negation of P 3. “P or Q” : (P \/ Q) disjunction of P and Q 4. “P implies Q” : ( P  Q) implication • P : antecedent, premise, hypotheses; • Q: consequence, conclusion • equivalent phrase: • “P entails Q”, “Q if P”, “P only if Q” 5. “p iff q” (p <->q) equivalence of P and Q

  7. Truth conditions for compound propositions 1. P /\ Q is true iff both P and Q are true 2. P \/ Q is true iff either P or Q is true 3. ~P is true iff P is not true (i.e., false) 4. P<->Q is true iff both are true or both are false. 5. PQ is true iff whenever P is true, Q is true ( i.e., it is not the case that P is true but Q is false ) • In the propositions, /\, \/, <-> and  are called binary connectives(or operations) and ~ is called unary connective(or operations). • Other binary connectives are possible. • What about the truth condition for primitive propositions?

  8. Natural language statement Today is Friday I have a test today It is not the case that today is Friday Today is not Friday Today is Friday and I have a test today If today is Friday, then I have a test today. Today is Friday or I have a test today. symbolized logic expr. p q ~p ~p (P ∧ q) (pq) (P ∨ q) From natural language statements to formal logic expressions

  9. Truth conditions for propositions • Problem: when will you say that the sentence: It_is_raining is true ? => The proposition:” It_is_raining” is true if the meaning (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation that the sentence referred to. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, then I would say that It_is_raining is true. • Factors affecting the truth value of a proposition: • the situation in which the proposition is used. • the meaning of the proposition.

  10. Truth table • The meaning of logical connectives can be • represented by a truth table.

  11. Other usual connectives • Besides /\,\/, and <->, there are also some other connectives as you have known in digital system. • xor , nand, nor. • Rules: let I(x) denote the truth value of x. • I(A  B)= I(~(A<->B))=1 iff not I(A)=I(B) • I(A nand B) = I(~(A/\B)). • I(A nor B) = I(~( A \/B)).

  12. 1.2 Propositional Equivalence • Some definitions: Let A be any proposition, then 1. A is a tautology(contradiction) iff A is true(false) no matter what the truth values of the prop. variables inside A are given. 2. If A is neither a tautology nor a contradiction, then A is called a contingency. Contingencies Contradictions (unsatisfiable) Tautologies (valid) The geography of propositions

  13. Logical Equivalence • A and B are logically equivalent (,A  B) if A <-> B is a tautology • Theorem1: Logical equivalence relation is reflexive, symmetric and transitive. I.e., for all propositions A,B and C, 1. A  A 2. A  B implies B A and 3. A  B and B  C imply A  C. • Theorem2[substitution theorem]: If A  B and C[X] is a proposition containing X as a subproposition, then C[A] and C[B] are logically equivalent, where C[A] is the result of C with X in C replaced by A. • ex: (p∨q)  (q∨p), C[X] =def ~(p ∧ X) • => ~(p∧ (p∨q) )~(p∧ (q∨p))

  14. Determine tautologies, contradictions and contingencies • Problems: Given a proposition A, how can you determine whether it is a tautology, a contradiction or a contingency? =>Exhausted evaluation! (by using truth table) e.g., A = (p->q) <-> (~p \/q) a tautology ? 1. there are two primitive props: p, q. 2. So there are 4 possible assignments of truth values to p and q. p=0,q=0: p=0,q=1: p=1,q=0: p=1,q=1: 3. Under each assignment, we can evaluate the truth value of A (in bottom up way) by the truth condition rule of connecting connectives.

  15. (pq)<->~p\/q :1 • p->q:1 ~p \/q:1 ~p:1 q:0 p:0 The evaluation tree for (p->q)<-> (~p\/q) 1 1 1 1 1 0 0 0 1 0 1 1 The truth table for pq <-> ~p\/q Determine tautologies A tautology since all are 1’s

  16. Determine other properties 1. logical equivalence: A <=> B iff A<->B is a tautology. So we can apply the procedure for determining tautology to determine if A <=>B. 2. contradiction: A is a contradiction iff ~A is a tautology 3. Satisfiability: A is satisfiable iff ~A is not a tautology. 4. contingence: A is a contingence iff neither A nor ~A is a tautology. 5. logical consequence:B is a logical consequence of A [denoted A |= B] iff AB is a tautology.

  17. Some important logical equivalences • P /\ T  P P/\P  P • P \/ F  P P\/P P • P \/ T  • P /\ F  • ~(~P)  • P \/ Q  • P /\ Q  • P \/ (Q \/ R)  • P /\ (Q /\ R)  • ==> P /\ (Q /\ R) = (P/\Q/\R)  • P /\(Q \/ R)  • P \/(Q /\ R)  • ~(P /\Q)  • ~(P\/Q)  ==> ~(P1 /\ P1/\.../\Pn)  • Identity law • Idempotent law • Domination law • Double negation • Commutative • Associative • Distribution law • DeMorgan’s law

  18. One Example: • Show that ~(p \/ (~p /\ q)  ~p /\ ~q Proof: 1. By truth table (omitted) 2. ~(p \/ (~p /\ q)  deM ~p /\ ~(~p /\ q)  deM ~p /\ (~(~p) \/ ~q)  DbNeg ~p /\ (p \/ ~q)  Distr (~p \/ ~q) /\ (~p \/ p)  ExMd (~p \/ ~q) \/ T  identity (~p \/ ~q)

  19. 1.3 Predicates (述詞) and quantifiers(量詞) • Problems of Propositional logic: • Cannot analyze the content of primitive propositions. • cannot group similar propositions • cannot analyze quantifiers. Ex: Consider the following situations: • 1. Family = {a,b,c1,c2,c3 } • 2. We are interested in who is whose brother. • 3. Suppose In Family, ci is cj’s brother if i ¹ j for i,j =1..3.

  20. Inadequancy of propositional logic • Primitive propositions needed: • x_is_y’s_brother, x,y Î Family • So we need 25 propositions. • New approach: Parameterized proposition(Predicate). • Use a (predicate) symbol B to symbolize the statement: • B(x,y) º x is y ‘s brother. • e.g., • B(a,c1) : a is c1’s brother, • B(c2,c2) : c2 is c2’s brother, • ...

  21. Benefits of Predicates • Advantage of using predicates: ==> 1. need only 6 (1 predicate and 5 names) symbols. 2. can group related propositions together. (B(x,y) and others, for instance, q(x) º x_is_old. ) 3.The most important: allow representation of quantified statement like: c1_has_a_brother (or, there is an object x which is c1’s brother, $ x B(x, c1)) to be inferred from c2_is_c1’s brother ( B(c2,c1) )

  22. Predicates • In the expression: B(x,y ) • B : predicate symbol (or proposition function) • x (y) : first (2nd) argument • B(x,y) : [atomic] proposition(or formula) • #arguments in B(X,Y): the arity of the predicate symbol B. • Notes: 1. Each arity has a fixed arity. 2. B(x,y,c1) and B(x) are meaningless. because used #arguments different from B’s arity.

  23. Predicates (cont’d) 3. The truth value of a proposition p(x1,...,xn) depends on both p and x1,...,xn. Ex1: Let p(x) º “x >3 “. then • P(4) º “4 > 3” has truth value T, • P(2) º “2 > 3” has truth value F. Ex2: Let Q(x,y) º “x = y +3” then • Q(1,2) = “1 = 2 + 3” is false. • Q(3,0) = “3 = 0 + 3” is true. • Mathematically, we can define (the meaning of) P as a function [P] • [P]: Un (n is P’s arity) --> {T,F}with the intention that • “P(x1,...,xn)” holds (is true) iff [P]([x1],...,[xn]) = T.

  24. Set representations of truth-functions • Ex: • [B]: Family2 -->{T,F} s.t. [b](x,y) = T iff x =[ci], y=[cj], i ¹ j. • p(x) º “x >3 “, So [P]: N --> {T,F } s.t. [P](x) = T iff x > 3. • Q(x,y) º “x = y +3” , so • [Q] : N2 --> {T,F} with • [Q](x,y) = T iff x = y +3. Note: P partition N into two sets, one with members mapped to T and one mapped to F. So, instead of represent P as a truth function, we equivalently represent P as the set {x | P(x) = T } = {4,5,6,...} P(x) = T. {1,2,3} {4,5,...} P(x) = F Similarly, we can represent Q as {(x,y) | x = y + 3 } N

  25. constant, function, variables and terms • In the statement: “y > min(x, 3)” --- (s1) • x,y are called variables, • 3 is a constant, • min is a function symbol with arity 2 • “min(3,2)” behaves more like x, 3 than “x >y”. • So if let P(x,y) º “x > y”, then • s1 can be represented as P(y, min(x,3)) • we call any expression that can be put on the argument position of an atomic proposition a term • Obviously, constants and variables are terms; moreover,if f is a function of arity n, and t1,...tn are n terms, then so is f(t1,...,tn).

  26. Universal and existential quantifications • A: any statement (or called formula) • e.g., A = p(x,y) /\ q(x,f(3)),... x: any variable, Then 1. ("x A) is a new formula called the universal quantification of A, 2. ("x A) means: “A(x) is true for all values of x in the universe of discourse.” 3. ($x A) is a new formula called the existential quantification of A, 4. ($x A) means: “A(x) is true for some value of x in the universe of discourse.”

  27. Quantifiers and universe of discourse • Consider the sentence: p(x) = “x > 0”. • 1. as stated before, p(x) is true or false depending on the value of x. • 2. p(1), p(2),... : true; p(0) , p(-1),...: false • Universe of Discourse (U,論域): • The set of objects (domain) from which all variables are allowed to have values. • Ex: consider the sentence: s2: ”all numbers are greater than 0”. (for-all x “x > 0” ) or (" x p(x) ) • U = N+={1,2,3,...} => s2 is true. U = Z={...,-1,0,1,...}=> s2 is false.

  28. Translation of NL statement into logical one • Translate the following sentences into logical expressions: 1. Every body likes Chang 2. Everybody loves somebody. 3. There is somebody loved by every one. 4. Nobody likes everybody. 5.There is somebody Chang don’t like. 6. there is somebody no one likes. 7.there is exactly one person everybody likes. 8 Chang likes exactly two persons 9. Everyone likes himself. 10. There is someone who likes no one who does not like himself.

  29. Terminology about formulas • Occurrences of variables in formulas: • A = p(f(x), c) \/ q(x,y) • A contains 2 variables, one of which occurs twice. • Free and bound variables: • A: a formula • fv(A) = the set of all free variables in A is defined as follows: • If A is atomic => fv(A) is the set of all variables in A • fv(~A) = fv(A) • fv(A/\B)=fv(A\/B)=fv(A->B)=fv(A<->B)= fv(A) U fv(B). • fv( for-all x A) = fv($xA) = fv(A)\ {x}.

  30. bd(A): the set of bound variables in A is defined as follows; • 1. If A is atomic => bd(A) = {} • 2. bd(~A) = bd(A) • 3. bd(A/\B) = d(A\/B) = bd(AB) =bd(A<->B) = bd(A) U bd(B). • 4. bd( ∀x A) = • bd(A) if x ∉ fv(A). bd(A) U {x} if x ∈ fv(A). • Example A = ($ x p(x,y)) \/ q(x) => fv(A) = {x,y}; bd(A) = {x} x occurs twice in A, one occurs free and the other occurs bound. • Def: A is a sentence iff fv(A) = {} • Note: A is a proposition if A is a sentence

  31. More logical notions [omitted!!] • A  B [logical equivalence (in predicate logic)] iff A <-> B is a tautology(or valid) for all possible values of free variables in A and in B. • Common equivalences: 1. all propositional equivalences • A->B  ~A \/ B, ~~A  A, ~(A/\B)  ~A\/~B,... 2. ~ "x A < $x ~A. 3. ~$xA "x ~A. 4. "x (A /\ B) "x A /\ "xB. 5. $x(A \/B) $xA \/ $XB If x is not free in B, and Q is " or $, then 6 Qx (A /\ B)  QxA /\ QxB 7. Qx(A\/B)  QxA \/ QxB

  32. 1.4 Sets • Basic structure upon which all other (discrete and continuous ) structures are built. • a set is a collection of objects. • an object is anything of interest, maybe itself a set. • Definition 1. • A set is a collection of objects. • The objects is a set are called the elements or members of the set. • If x is a member of a set S, we say S contains x. • notation: x ÎS vs x Ï S • Ex: In 1,2,3,4,5, the collection of 1,3 5 is a set.

  33. Set description • How to describe a set:? 1. List all its member. • the set of all positive odd integer >10 = ? • The set all decimal digits = ? • the set of all upper case English letters = ? • The set of all nonnegative integers = ? 2. Set builder notation: • P(x) : a property (or a statement or a proposition) about objects. e.g., P(x) = “ x > 0 and x is odd” • then {x | P(x) } is the set of objects satisfying property P. • P(3) is true => 3 Î {x | P(x)} • P(2) is false => 2 Ï {x | P(x)}

  34. Conventions • N =def {x | x is a natural numbers } = { 0,1,2,3,...} • N+ =def {1,2,3,...} • Z =def {...,-3,-2,-1,0,1,2,3,...} • R =def the set of real numbers. • Problem: The same set may have many different descriptions. • {x | 0 < x <10 /\ x is odd } • {1,3,5,7,9}, {5,3,1,9,7} • {9,7,1,3,5}.

  35. Set predicates Definition 2. • Two sets S1, S2 are equal iff they have the same elements • S1 = S2 iff "x (x Î S1 <-> x Î S2) • Ex: {1,3,5} = {1,5,3} = {1,1,3,3, 5} • Graphical representation of sets: • Venn Diagrams: • rectangle: Universal set: U • circles: sets • points: particular elements • point x inside circle S => x Î S • point x outside S => x Ï S.

  36. Set predicates (cont’d) • Null set ={} = Æ =def the collection of no objects. Def 3’: [empty set] for-all x x Ï Æ. Def 3. [subset] • A Í B iff all elements of A are elements of B. • A Í B  for-all x (x Î A ⇒ x Î B)). Def 3’’: A Ì B =def A Í B /\ A ¹ B. • Exercise : Show that: • 1. For all set A (Æ Í A) • 2. (A Í B /\ B Í A)  (A = B) • 3. A Í Æ ⇒ A = Æ • Diagram representation of the set inclusion relationship.

  37. Size or cardinality of a set Def. 4: | A | = the size(cardinality) of A = # of distinct elements of A. • Ex: • |{1,3,3,5}| = ? |{}| = ? • | the set of binary digits } | = ? • |N| = ? ; |Z| = ? ; | {2i | i ∈ N} = ? • |R| = ? Def. 5. • A set A is finite iff |A| is a natural number ; o/w it is infinite. • Two sets are of the same size (cardinality) iff there is a 1-1 & onto mapping between them.

  38. countability of sets Exercise: Show that • 1. |N| = |Z | = | Q | = |{4,5,6,...}| • 2. |R| = | (-1, 1) | = |(0,1)| • 3. |(0,1)| is uncountable Def. • A set A is said to be denumerable iff |A| = |N|. • A set is countable iff either |A| = n for some n in N or |A| = |N|. By exercise 1,2,3, • R is not countable. • Q and Z is countable.

  39. The power set Def 6. • If A is a set, then the collection of all subsets of A is also a set, called the poser set of A and is denoted as P(A) or 2A. • Ex: • P({0,1,2}) = ? • P({}) = ? • |P({1,2,..., n})| = ? • Order of elements in a set are indistinguishable. But some sometimes we need to distinguish between (1,3,4) and (3,4,1) --> ordered n-tuples

  40. Show that |A| ≠ |2A| Pf: (1) The case that A is finite is trivial since |2A| = 2|A| > |A| and there is no bijection b/t two finite sets with different sizes. (2) assume |A| = |2A|, i.e., there is a bijection f: A -> 2A. Let D = {x ∈ A | x Ï f(x) }. ==> 1. D is a subset of A; Hence 2. $y ∈ A s.t. f(y) = D. Problem: Is y Î D ? if yes (i.e., y Î D) ==> y Ï f(y) = D, a contradiction if no (i.e., y Ï D) ==> y Î f(y) =D, a contradiction too. So the assumption is false, i.e., there is no bijection b/t A and 2A.

  41. P HALT(P,X) Pr(x:String) { ….. } yes P(X) Halt ? true no false “It’s a test …” X The Halting Problem • L : any of your favorite programming languages (C, C++, Java, BASIC, etc. ) • Problem: write an L-program HALT(P,X), which takes another L-program P(-) and string X as input, and HALT(P,X) returns true if P(X) halts and returns false if P(X) does not halt. Program you are asked to design

  42. Halt(P,X) does not exist • Motivations of the proof: Problem1 : What about the truth value of the sentence: L: L is false Problem 2 : Let S = {X | X X}. Does S belong to S or not ? The analysis: S  S => S  S; S  S => S  S. Conclusion: • 1. S is not a set!! • 2. If a language is too powerful, it may produce expressions that is meaningless or can not be realized. • Question: If HALT(P,X) can be programmed, will it incur any absurd result like the case of S? Ans: yes!!

  43. Loop P HALT(P,X) Pr(x:String) { ….. } yes P(P) Halt ? true END no false H(P) : Another program using HALT(-,-) as a subroutine H(P) P P Note: H(P) Halts iff HALT(P,P) returns false iff P(P) does not halt.

  44. H(P) Loop HALT(P,X) HALT(P,X) yes yes P(X) Halt ? P(X) Halt ? true true no no END false false Problem: Will H(H(P)) Halt? H(H(P)) Loop END H(P) The anomaly : H(H(P) halts iff H(H(P)) does not halt.

  45. Cartesian Products Def. 7 [n-tuple] • If a1,a2,...,an (n > 0) are n objects, then “(a1,a2,...,an)” is a new object, called an (ordered) n-tuple [ with ai as its ith element. ] • Any ordered 2-tuple is called a pair. • (a1,a2,...,am) = (b1,b2,...,bn) iff • (1) m = n and (2) for i = 1,..,n ai = bi.

  46. Cartesian product Def. 8: [Cartesian product] A x B =def {(a,b) | a ∈ A and b ∈ B } A1 x A2 x ...x An =def {(a1,...,an) | ai ∈ Ai }. Ex: A = {1,2}, B = {a,b,c} , C = {0,1} 1. A x B = ? ; 2. B x A = ? 3. A x {} = ? ;4. A x B x C = ? problem: 1. when will A x B = B x A ? 2. |A x B | = ?

  47. 1.5 Set operations • union, intersection,difference , complement, • Definition. 1. AÈ B = {x | x ∈ A or x ∈ B } 2. AÇ B = {x | x ∈ A and x ∈ B } 3. If A Ç B = {} => call A and B disjoint. 4. A - B = {x | x ∈ A but x ∉ B } 5. ~ A = U - A • Venn diagram representations • Ex: U = {1,..,10}, A = { 1,2,3,5,8} B = {2,4,6,8,10} => AÈ B , AÇ B , A - B , ~ A = ?

  48. Set identities • Identity laws: A ? ? = A • Domination law: U ? ? = A; {} ?? = {} • Idempotent law: A ? A = A ; • complementation: ~~A = A • commutative : A ? B = B ? A • Associative: A ? (B ? C) = (A ? B ) ? C • Distributive: A ? (B ? C) = ? • DeMoregan laws: ~(AUB) = ?; ~(A ⋂ B)=?

  49. Prove set equality 1. Show that ~(A È B) = ~A Ç ~B by show that • 1. ~(A È B) Í ~A Ç~B • 2. ~A Ç~B Í ~(A È B) pf: (By definition)Let x be any element in ~(A È B) ... 2. show (1) by using set builder and logical equivalence. 3. Show distributive law by using membership table. 4. show ~(AÈ (BÇ C)) = (~C Ç~B) È ~A by set identities. Note the similarity between logical equivalence and set identities.

  50. Generalized set operations Def. 6 • A1,A2,...An: n sets • B = {A1,A2,...An } 1. A1È A2È An = ÈB = È{i=1,..n} Ai =def ? 2. A1Ç A2Ç An = ÇB = Ç {i=1,..n} Ai=def ? quiz: if B = {} => ÈB = ?; ÇB = ? • Venn diagram representation of A1È A2È A3 and A1Ç A2Ç A3 . Example: A = {0,2,4,6,8}, B = {0,1,2,3,4}, C={0,3,6,9} => • A1È A2È A3 = ? • A1Ç A2Ç A3 = ?

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