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Agenda • Review Minggu 1: – Partition • Topik Minggu 2: – Propositions – Conditional Propositions & Logical Equivalence – Arguments & Rules of Inference – Quantifiers – Nested Quantifiers • Latihan Soal Review: Partition • A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets X is a disjoint union of the subsets. • A family of sets P is a partition of X if and only if all of the following conditions hold: – P does not contain the empty set. – The union of the sets in P is equal to X. (The sets in P are said to cover X.) – The intersection of any two distinct sets in P is empty. (We say the elements of P are pairwise disjoint . • Propositions Propositions/Statements • A statement (or proposition) is a sentence that is true or false but not both. • The truth value of a proposition is either TRUE / T / 1 or FALSE / F / 0. • Ex. – two plus two equals four • Proposition? Yes • Truth value: true – Jakarta is the capital of Singapore • Proposition? Yes • Truth value: false Examples • Two plus two equals five – Proposition? Yes – Truth value: False • An elephant is bigger than an ant – Proposition? Yes – Truth value: true • He is a university student – Proposition? No – Truth value: depends on who he is • C is bigger than 10 – Proposition? No – Truth value: unknown • F plus G equals 9 – Proposition? No – Truth value: unknown Examples: • Dimana letak kampus UMN? – Proposition? No (pertanyaan) • Jangan memakai sandal ke kampus – Proposition? No (perintah) • Mudah-mudahan jalan tidak macet – Proposition? No (harapan) • Indahnya bulan purnama – Proposition? No (ketakjuban / keheranan) Compound Propositions / Compound Statements • A composition of two or more propositions / statement that is true or false but not both • Example: – Budi is studying at UMN, he is a university student • Compound statement? Yes • Truth value : True – Jika x = 1 dan y = 2 maka x lebih besar daripada y • Compound Statement? Yes • Truth value: False Examples • x ≤ a means x < a or x = a • a ≤ x ≤ b means a ≤ x and x ≤ b • 2≤x≤1 – compound statement? Yes – Truth value: False Formalization of (Compound) Statements • Translating a (compound) statement to symbols (such as x, y, z) and logical operator. • Logical operator: ~,¬ not and or Examples ¬p : not p, negation of p p q : p and q, conjunction of p and q p q : p or q, disjunction of p and q • Order of operation : (… ) ~, ¬ Example: ¬p q = (¬p) q p q r Is it (p q) r or p (q r) ? To be safe, use (…) Examples • p: Today is Friday • Negation: ~p: Today is not Friday • p: At least 10 inches of rain fell today in Jakarta • Negation: ~p: Less than 10 inches of rain fell today in Jakarta Examples • p = it is hot; q = it is sunny • It is not hot but sunny – It is not hot and it is sunny ~p q • It is neither hot nor sunny – It is not hot and it is not sunny ~p ~q Truth Table The list of all possible truth values of a compound statement. Truth Table for Negation Truth Table for Conjunction p q It is hot and it is sunny Truth Table for Disjunction p q It is hot or it is sunny Truth Table for Exclusive Or It is hot or it is sunny, but not both Definition: (p q) ~(p q) : p q, p XOR q, Evaluating the Truth of more General Compound Statements ~p q = (~p) q Steps: - Evaluate the expressions within the innermost parentheses - Evaluate the expressions within the next innermost set of parentheses - Until you have the truth values for the complete expression. Evaluating the Truth of more General Compound Statements p T T F F q T F T F ~p F F T T ~p q F F T F Tautology and Contradiction Tautology: True (for any truth values of their variables) Contradiction: False (for any truth values of their variables) Contoh: Tautology Contradiction Notes on Programming Language p q = p && q p q = p || q ~p = !p Conditional Propositions & Logical Equivalence Conditional Proposition Definition • Let p and q be propositions. The conditional proposition p q is the proposition “if p then q”. p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence). • “p implies q” “p q”; p: hypothesis, q: conclusion. • Conditional: the truth of statement q is conditioned on the truth of statement p • Example: IF 36 is divisible by 6, THEN 36 is divisible by 3 Conditional Proposition • IF Maria learns discrete mathematics, THEN she will find a good job. • p: Maria learns discrete mathematics • q: she will find a good job. • pq • Under what circumstances is the above sentence false? • False when Maria learns discrete mathematics but not find a good job • IF you show up for work Monday morning, THEN you will get the job. • Under what circumstances is the above sentence false? Truth Table for Conditional Proposition Definition • p q is false when p is true and q is false; otherwise it is true. Example: Biconditional Proposition Definition • Given statement variables p and q, the biconditional of p and q is “ p if and only if q” and is denoted p q. The words if and only if are sometimes abbreviated iff. Priority • Logical operator: () ~, ¬ , not and or if-then, iff Logical Equivalence Definition: • Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variable. P=pq Q=qp • The logical equivalence of statement forms P and Q is denoted by writing P Q ~(~p) p Are ~(p q) and ~p ~q logically equivalent? De Morgan’s Laws Definition: • The negation of an AND statement is logically equivalent to the OR statement in which each component is negated. ~(p q) ~p ~q • The negation of an OR statement is logically equivalent to the AND statement in which each component is negated. ~(p q) ~p ~q De Morgan’s Laws: Truth Table De Morgan’s Laws: Exercise Use De Morgan’s Laws to find the negation of each of the following statements: • Jan is rich and happy • Carlos will bicycle or run tomorrow • Melani walks or takes the bus to class • Ibrahim is smart and hard working Representation of IF-THEN as OR • p: you do not get to work on time • q: you are fired • IF you do not get to work on time THEN you are fired • ~p: you get to work on time • You get to work on time OR you are fired p q ~p q Negation, Contrapositive, Converse, Inverse The Negation of a Conditional Proposition • The negation of “IF p THEN q” is logically equivalent to “p and not q” ~(p q) p ~q • Show the equivalence by using Morgan Law: ~(p q) ~(~p q) ~(~p) ~q p ~q The Negation of a Conditional Proposition • Exercise: Truth table for ~(p q) p ~q The Negation of a Conditional Proposition • ~(IF my car is in the repair shop, THEN I cannot get the class) • My car is in the repair shop and I can get to class • ~(IF Sara lives in Jakarta, THEN she lives in Indonesia) • Sara lives in Jakarta and she does not live in Indonesia Biconditional Proposition • Is “ p if, and only if, q” logically equivalent with “ if p then q “ and “if q then p” ? p q (p q) (q p) Biconditional Proposition: Truth Table The Contrapositive of a Conditional Proposition Definition • The contrapositive of a conditional statement of the form “IF p THEN q” is “IF ~q THEN ~p” • The contrapositive of p q is ~q ~p • Are they logically equivalent? Construct the truth table • A conditional statement is logically equivalent to its contrapositive. Contrapositive: Examples • IF Howard can swim across the lake, THEN Howard can swim to the island • IF Howard cannot swim to the island, THEN Howard cannot swim across the lake • IF today is Easter, THEN tomorrow is Monday • IF tomorrow is not Monday, THEN today is not Easter The Converse and Inverse of a Conditional Statement Definition • Suppose a conditional statement of the form “IF p THEN q” is given. • The converse is “IF q THEN p” • The inverse is “IF ~p THEN ~q” Symbolically • The converse of p q is q p • The inverse of p q is ~p ~q Are they logically equivalent? Construct the truth table Converse and Inverse: Truth Table p q ~p ~q p q q p ~p ~q T T F F T T T T F F T F T T F T T F T F F F F T T T T T Converse and Inverse: Examples • IF Howard can swim across the lake, THEN Howard can swim to the island • Converse: IF Howard can swim to the island THEN Howard can swim across the lake • Inverse: IF Howard cannot swim across the lake, THEN Howard cannot swim to the island. • IF today is Easter, THEN tomorrow is Monday • Converse: IF tomorrow is Monday, THEN today is Easter • Inverse: IF today is not Easter, THEN tomorrow is not Monday Arguments & Rules of Inference Argument The bug is either in module 17 or in module 81 The bug is a numerical error Module 81 has no numerical error Conclusion: The bug is module 17 • Deductive reasoning: drawing a conclusion from a sequence of propositions. Argument Definition • A (deductive) argument is a sequence of hypotheses that ends with a conclusion. IF premise-1, ….., premise-n THEN conclusion • An argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false. If the premises are all true, then the conclusion is also true, otherwise the argument is invalid. Argument premise-1 premise-2 . . . premise-n q • The symbol , read “therefore” Example • If you have a current password, then you can log onto the network • You have a current password • Therefore, • You can log onto the network pq p q • Construct the truth table p q, p q Testing the Validity of an Argument • Identify the premises and conclusion of the argument • Construct a truth table showing the truth values of all the premises and the conclusion • Find the rows (called critical rows) in which all the premises are true • In each critical row, determine whether the conclusion of the argument is also true. – If in each critical row the conclusion is also true, then the argument form is valid – If there is at least one critical row in which the conclusion is false, the argument form is invalid Valid or Invalid Argument ? p ( q r) ~r p q Valid or Invalid Argument ? p q ~r qpr p r Modus Ponens pq p q Modus ponens: method of affirming Modus Tolens pq ~q ~p Exercise: Construct the truth table Modus tolens: method of denying Modus Tolens Example It is below freezing now Therefore, It is either below freezing or raining now p pq Valid argument : addition rule Example It is below freezing and raining now Therefore, It is below freezing now pq p Valid argument : simplification rule Example If it rains today, then we will not have a barbecue today If we do not have a barbecue today, then we will have a barbecue tomorrow Therefore, If it rains today, then we will have a barbecue tomorrow pq q r p r Valid argument : hypothetical syllogism Rule of Inference and Arguments To show whether an argument is valid, when there are many premises in an argument: - Construct truth table (not efficient) - Use several rules of inference Example It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny If we do not go swimming, then we will take a canoe trip If we take a canoe trip, then we will be home by sunset Conclusion: We will be home by sunset Quantifiers How to express? Can we express the following statements by using propositional logic we have learned before? “Every computer connected to the university network is functioning properly” “computer-1 is functioning properly” “There is a computer on the university network that is under attack by an intruder” No! Introducing Predicate Logic Predicates Let P(x) be a statement involving the variable x and let D be a set. We call P a propositional function or predicate (with respect to D) if for each x D, P(x) is a proposition. D is the domain of discourse of P. Predicates “x is greater than 3” x: variable “greater than 3”: the property that the variable can have. “x is greater than 3”: propositional function (predicate) P at x or P(x) Once a value has been assigned to x, the statement P(x) becomes a proposition and has a truth value. P(x) : (x > 3) Truth values of P(4) and P(2)? P(4) is true, P(2) is false Predicates A(x) : Computer x is under attack by an intruder Reality: comp-1 and comp-9 are under attack A(comp-1) and A(comp-9) are true, but A(comp-5) is false. Q(x,y) : x2 > y2 + 6 Q(3,2)? Q(4,3)? Quantifiers Quantification expresses the extent to which a predicate is true over range of elements (domain). Ex. all, some, many, none, few. Predicate calculus: the area of logic that deals with predicates and quantifiers. Universally Quantified Statement: a predicate is true for every element under consideration. Existensially Quantified Statement: there is one or more element under consideration for which the predicate is true. Universal Quantifiers • Definition “P(x) for all values of x in the domain” Where P(x) is statement. • Notation: x P(x) For all x P(x), or for every x P(x), or for any x P(x) • Counterexample: an element of x for which P(x) is false. Example • • • • P(x) : x + 1 > x Domain: all real numbers. What is the truth value of x P(x)? True, because P(x) is true for all real numbers. • • • • Q(x) : x < 2 Domain: all real numbers. x Q(x)? Counterexample: x = 3; x Q(x) is false. Example • • • • P(x) : x2 > 0 Domain: all integers. x P(x)? Counterexample: x = 0; x P(x) is false. • • • • Q(x) : x2 < 10 Domain: positive integers not exceeding 4 x Q(x)? Counterexample: x = 4; x Q(x) is false. Existensial Quantifier • Definition “There exists an element x in the domain such that P(x)” Where P(x) is a statement. • Notation: x P(x) there is an x such that P(x), or there is at least one x such that P(x), or for some x P(x) Existensial Quantifier • Definition “There exists an element x in the domain such that P(x)” Where P(x) is a statement. • Notation: x P(x) there is an x such that P(x), or there is at least one x such that P(x), or for some x P(x) Example • • • • P(x) : x > 3 Domain: all real numbers. What is the truth value of x P(x)? True, because P(x) is true for x = 4. • • • • Q(x): x = x + 1 Domain: all real numbers. x Q(x)? False, because Q(x) is false for every real number x. Example • • • • Q(x): x2 > 10 Domain: positive integer not exceeding 4. x Q(x)? True, because Q(x) is true for x = 4. Summary Statement When True? When False? x P(x) P(x) is true for every x x P(x) There is an x for P(x) is false for which P(x) is true every x There is an x for which P(x) is false Negation • “every student in your class has taken a course in calculus” • P(x): x has taken a course in calculus • Domain: the students in your class • x P(x) • Negation: • “there is a student in your class who has not taken a course in calculus” • x P(x) x P(x) • x Q(x) x Q(x) Generalized De Morgan’s Laws for Logic Negation Equivalent x P(x) x P(x) x P(x) x P(x) When is Negation True? For every x, P(x) is false When False? There is an x for which P(x) is true There is an x for which P(x) is true P(x) is false for every x Nested Quantifiers Nested Quantifiers How to express: The sum of any two positive real numbers is positive. Variables: x, y If x > 0 and y > 0 then x + y > 0 OR P(x, y): (x > 0) (y > 0) (x + y > 0) x y P(x, y) Domain of discourse R x R In words: For every x and for every y, if x > 0 and y > 0, then x + y>0 Nested Quantifiers Hx y P(x, y) is true if for every x X and y Y, P(x, y) is true x y P(x, y) is false if there is at least one x X and at least one y Y, P(x,y) is false counterexample x y (x > 0) (y < 0) (x + y ≠ 0) ? Counterexample: x = 1 and y = -1 ow to express: In words: For every x and for every y, if x > 0 and y > 0, then x + y>0 Nested Quantifiers How to express: Everybody loves somebody. L(x, y): x loves y “everybody”: universal quantification “somebody” existensial quantification x y L(x, y) In words: For every person x, there exists a person y such that x loves y This is not a correct interpretation: x y L(x, y) In words: There exists a person x such that for all y, x loves y Nested Quantifiers x y L(x, y) is true if for every x X, there is at least one y Y for which L(x,y) is true x y L(x, y) is false if there is at least one x X such that L(x, y) is false for every y Y x y (x + y = 0) Domain R x R True: for every real number x, there is at least one y (namely y = -x) for which x + y = 0 x y (x > y) Domain Z+ x Z+ False: there is at least one x, namely x = 1, such that x > y is false for every positive integer y. Nested Quantifiers x y L(x, y) is true if there is at least one x X such that L(x, y) is true for every y Y x y L(x, y) is false if for every x X, there is at least one y Y such that L(x,y) is false x y L(x, y) is true if there is at least one x X and at least one every y Y such that L(x, y) is true x y L(x, y) is false if for every x X and for every y Y, L(x,y) is false The Generalized Morgan’s law for logic ~(x y L(x, y)) = x ~(y L(x, y)) = x y ~L(x, y)
