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Unformatted text preview: The Logic of Compound Statements Mariusz Bajger COMP2781/8781 School of Computer Science, Engineering and Mathematics March 8, 2011 1/29 Reading and Exercises Reading Epp, Chapter 2, all sections except 2.5 Exercises Set 2.1 (all in blue), Set 2.2 (all in blue), Set 2.3 (all in blue), Set 2.4 (all in blue except 33, 34) Good learning strategy: BE ACTIVE! I Regularly revise lectures I Solve the suggested exercises I Be critical when reading textbook/lectures I Ask your colleagues, ask the lecturer, don’t be shy! I It is OK to ask for help any time 2/29 Statements (propositions) The following terms are not formally defined: sentence , true and false . Definition Statement or proposition is a sentence that is either true or false (but not both). Which sentences are statements/propositions? I 2 * 5 = 22 I Let me go! I Do you like maths? I If ab = 0 then either a = 0 or b = 0, where a , b ∈ R I 2 + 6 I x + 7 = 2 I Julia Gillard is popular. I This sentence is false. 3/29 Compound Statements Built from simple statements using logical connectives I ∼ not I ∨ or disjunction I ∧ and conjunction Example p : It is raining q : It is hot p ∨ ∼ q reads: It is raining or not hot. Formal (which we use) and commonly used (intuitive) logic differ! How does ’but’ translates? He is tall but he is not heavy. He tried but failed. Neither are prices increasing nor are interest rates steady. ’but’ translates to ∧ , ’neither p nor q ’ means ∼ p ∧ ∼ q 4/29 The order of operations I ∼ goes first then I either ∧ or ∨ (coequal) I statement p ∧ q ∨ r is ambiguous! I Solution: use parentheses I For example: ( p ∨ q ) ∧ r 5/29 Truth values Every statement must have well-defined truth values (either true or false). Definition A statement form is an expression made up of statement variables (such as p , q , r ) and logical connectives (such as ∼ , ∨ , ∧ ) that becomes a statement when actual statements are substituted for the component statement variables. Example Consider ( p ∨ q ) ∨ ( ∼ r ) as a statement form, where p : John is tall q : Jack is heavy r : It is raining 6/29 Evaluating truth values I ∼ p is true if p is false and is false if p is true I p ∧ q is true when, and only when, both p and q are true I p ∨ q is true when either p is true or q is true, or both p and q are true I The truth table for a statement form lists the truth values that correspond to all possible combinations of truth values for its component statement variables....
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