Lecture 1: logic Learning objectives: • The student be able to determine the propositional logic. • The student be able to understand the logical connectives. • The student be able to construct the truth tables. • The student will be able to validate the arguments. Proposition A proposition (or a statement) is a sentence that is true or false but not both. Example 1: 1. “Two plus two equals four” ā« – It is True 2. “Two plus two equals five” ā« – It is False BOTH ARE STATEMENTS Are the following propositions? 1. Sydney is the capital of Australia. 2. Do Tutorial 11 now. 3. Today is Wednesday. 4. x is a prime number. Compound Proposition • Three symbols that are used to build more complicated logical expressions out of the simplers ones. ∼ ∧ ∨ not and or ∼q p∧q p∨š negation of q (not q) p and q p or q Compound Statement • Create new (compound) propositions from existing ones (elementary proposition) • The five connectives: 1. 2. 3. 4. 5. Negation (not …) Conjunction (… and …) Disjunction (… or …) Implication (if … then …) Equivalence (… if and only if …) Hierarchy Among Logical Connectives Translating from English to Symbols: But and Neither-Nor • Write each of the following sentences symbolically, letting h = “It is hot” and s = “It is sunny.” • It is not hot but it is sunny. • It is neither hot nor sunny. Suppose that x is a particular real number. Let p, q, and r symbolize ′0 < š„′, ′x < 3′, and ′x = 3′, respectively. Write the following inequality symbolically: a. x ≤ 3 a. 0 < x < 3 a. 0 < x ≤ 3 Truth Values - Negation Definition If p is a statement variable, the negation of p is “not p” or “It is not the case that p” and is denoted by ∼ š. It has the opposite truth value from p: if p is true, ∼ š is false; if p is false, ∼ š is true. š ∼š T F F T Truth Values - Conjunction Definition If p and q are statement variables, the conjunction of p and q is “ p and q” , denoted š ∧ š. It is true when, and only when, both p and q are true. If either p or q is false, or if both are false, š ∧ š is false. š š š∧š T T T T F F F T F F F F Truth Values - Disjunction Definition If p and q are statement variables, the disjunction of p and q is “ p or q” , denoted š ∨ š. It is true when either p is true, or q is true, or both p and q are true; it is false only when both p and q are false. š š š∨š T T T T F T F T T F F F Truth Values – Exclusive Or Truth Table for Exclusive Or If p and q are statement variables, the Exclusive Or means ‘p or q but not both” which translates into symbols as š ∨ š ∧ ~(š ∧ š). Also abbreviated as pāØš or š ššš š. š š š∨š š∧š ~ š∧š š ∨ š ∧ ~(š ∧ š) T T T T F F T F T F T T F T T F T T F F F F T F Logical Equivalence Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements variables. The logical equivalence of statement forms P and Q is denoted by writing š≡š Test Whether Two Statement Form P and Q are Logically Equivalent 1. Construct a truth table with one column for the truth table values of P and another column for the truth values of Q. 2. Check each combination of the statement variables to see whether the truth value of P is the same as the truth value of Q. Example 2: Show that the statement forms ~ equivalent š∧š and ~š ∨ ~š are logically š š ~š ~š š∧š ~ š∧š ~š ∨ ~š T T F F T F F T F F T F T T F T T F F T T F F T T F T T Example 2: Show that the statement forms ~ equivalent š∧š and ~š ∧ ~š are not logically š š ~š ~š š∧š ~ š∧š ~š ∧ ~š T T F F T F T F F T F T ≠ F F T T F F T ≠ F F F T T F T F T De Morgan’s Law De Morgan’s Laws The negation of an and statement is logically equivalent to the or statement in which each component is negated. The negation of an or statement is logically equivalent to the and statement in which each component is negated. ~ š ∧ š ≡ ~š ∨ ~š ~ š ∨ š ≡ ~š ∧ ~š De Morgan’s Laws ~ š ∨ š ≡ ~š ∧ ~š p q T T F T F F F ~š ~š š∨š ~(š ∨ š) ~š ∧ ~š F T F F F T T F F T T F T F F F T T F T T Applying De Morgan’s Laws • Write negations for each of the following statements: a. John is 6 feet tall and he weighs at least 200 pounds. b. The bus was late or Tom’s watch was slow. Solution: a. John is not 6 feet tall or he weighs less than 200 pounds. b. The bus was not late and Tom’s watch was not slow. Types of Compound Propositions Tautology • Always true! Contradiction • Always false! Contingency • Neither a tautology nor a contradiction Example 3: Show that the statement forms p ∨ ~š is a tautology and the statement form p ∧ ~š is a contradiction š ~š p ∨ ~š p ∧ ~š T F T F F T T F If t is a tautology and c is a contradiction, show that p ∧ š” ≡ š and p ∧ š ≡ š š š p∧š” p š p∧š T T T T F F F T F F F F Logical Equivalences Commutative Law Associative Laws Distributive Laws Identity Laws Negation Laws Double Negative Law Idempotent Laws Universal bound laws De Morgan’s Laws Absorption Laws Negations of t and c: Conditional Statements Definition If p and q are statement variables, the conditional of q by p is “If p then q” or “p implies q” and is denoted by š → š. • It is false when p is true and q is false; otherwise it is true. • We call p the hypothesis (or antecedent) of the conditional and q the conclusion (or consequent) š š š→š T T T T F F F T T F F T š → š ≡ ~p ∨ š Negation of Conditional Statements ~(š → š) ≡ ~(~š ∨ š) ≡ ~(~š) ∧ (~š) ≡ š ∧ ~š ~(š → š) ≡ š ∧ ~š Contrapositive of Conditional Statements Definition The contrapositive of a conditional statement of the form “If p then q” is If ~š then ~š Or symbolically, The contrapositive of š → š is ~š → ~š š → š ≡ ~š → ~š Converse and Inverse of Conditional Statements Definition Suppose a conditional statement of the form “If p then q” is given. 1. The converse is “ If q then p” 2. The inverse is “ If ~š then ~š.” Or symbolically, The converse of p → q is q → p, And, The inverse of š → š is ~š → ~š Both are not logically equivalent to š → š and to each other!!! Biconditional Statements Definition Given statement variable p and q, the biconditional of p and q is “p if, and only if, q” and is denoted š ↔ š. •It is true if both p and q have the same truth values. •It is false if p and q have opposite truth values. •Note: š ↔ š is a short form for (š → š) ∧ (š → š) •The word if and only if are sometimes abbreviated as iff. š š š↔š T T T T F F F T F F F T Example 4: Given that the truth values for propositions P, Q and R are T, F and T respectively. Determine the truth values of the following compound propositions: (a) P ļ Q ļ ~ R T ļ Fļ ~ T ļŗ T ļ F ļ F ļŗ F ļ F ļŗ F (b) ( P ļ® Q) ļ (Q ļ® R) (T ļ® F ) ļ ( F ļ® T ) ļŗ F ļ T ļŗ F (c) P ļ R ļ« Q ļ R T ļT ļ« F ļT ļŗ T ļ« T ļŗ T Only If Statements Definition If p and q are statements, p only if q means “if not q then not p” Or equivalently, “if p then q.” John will break the world’s record for the mile run only if he runs the mile in under four minutes. • If John does not run the mile in under four minutes, then he will not break the world’s record. • If John breaks the world’s record, then he will have run the mile in under four minutes. Something to ponder… p: It is a cat. q: It is an animal. Being an animal is a sufficient condition for being a cat. NO Being a cat is a sufficient condition for being an animal. YES š→š If it is a cat then it is an animal ~š →~p “p only if q” If it is not an animal then it is not a cat Being an animal is a necessary condition for being a cat. Valid and Invalid Arguments • An argument is a sequence of statements • All statements in an argument, except the final one, are called premises. • The final statement is called the conclusion. • The symbol ∴ which is read “therefore” is normally placed just before the conclusion. • To say that an argument form is valid means • if the premises are all true, • then the conclusion is also true. An Invalid Argument Form • Solution: T T T F T T T T T T T F T T F T F F T F T F F T F T T T F F T T F T T F F T T F T F T F T F T F T T F T F T F F T F F F T T T F F F T T F T T T Modus Ponens If p then q. p ∴q If 4,686 is divisible by 6, then 4,686 is divisible by 3. 4,686 is divisible by 6. ∴ 4,686 is divisible by 3 Modus Tollens If p then q. ~q ∴ ~p If Zeus is human, then Zeus is mortal. Zeus is not mortal. ∴ Zeus is not human. Valid Argument Forms --The End--