TAILIEUCHUNG - Lecture Formal methods in software engineering - Lecture 25

In this chapter, the following content will be discussed: Problem with truth table, proof using equivalences, rules for inference & logical deduction, logical problem for the day, associativity of equivalence, even and odd numbers, properties of equivalence,. | 1 1 Formal Methods in Software Engineering Lecture # 25 1 1 1 2 2 Problem with Truth Table Any problem with the truth table? Perform good when the number of proposition is small but if the number of proposition is large then? In truth table number of rows depends on the number of variables: If we have two variables then Another method is to built logical argument using deduction rules. 2 2 2 3 3 Some useful Equivalences 3 3 3 4 4 Some useful Equivalences 4 4 4 5 5 Proof using Equivalences A ( A => H ) A ( not A or H) (A and ( not A or H )) or (not A and not (not A or H)) A and ( not A or H) (A and not A ) or ( A and H) – Distributive law False or (A and H) A and H not A and not ( not A or H) Demorgans law not A and (A and not H) Associative (not A and A ) and not H (Negation Law) False and not H false 5 5 5 6 6 Proof using Equivalences Hence proved A ( not A or H) (A and H ) or false A and H 6 6 6 7 7 Rules for Inference & Logical Deduction Introduction Rules If two proposition is given then; p q (given) p and q (conclude) If one proposition is given then; p q if p is true then p and q is true p and q p and q Another; q if q is true then p=> will always true; p(not matter) p => q 7 7 7 8 8 Rules for Inference & Logical Deduction Elimination Rules p and q p and q if p and q is true then p is true p q if p is true and p=>q is also true so q has to be true p p => q q if your assumption is false then you can concluded anything p not p false false p 8 8 8 9 9 Rules for Inference & Logical Deduction Rules Importance Plays important role in logic . Contradiction If your assumption is false than no matter what ever you concluded . Implication (Pre-condition) 9 9 9 10 10 Rules for Inference & Logical Deduction Does the superman exist? If superman were able and willing to prevent evil , he would do so. If superman were unable to prevent evil, he would be incapable; if he were unwilling to prevent evil, he would be malevolent. Superman does not prevent evil. If

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