CSL105: Discrete Mathematical Structures. Ragesh Jaiswal, CSE, IIT Delhi
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3 is another way of showing that an argument is correct. Definitions: Literal: A variable or a negation of a variable is called a literal. Sum and Product: A disjunction of literals is called a sum and a conjunction of literals is called a product. Clause: A disjunction of literals is called a clause. Resolvent: For any two clauses C 1 and C 2, if there is a literal L 1 in C 1 that is complementary to literal L 2 in C 2, then delete L 1 and L 2 from C 1 and C 2 respectively and construct the disjunction of the remaining clauses. The constructed clause is a resolvent of C 1 and C 2. C 1 = P Q R C 2 = P S T What is a resolvent of C 1 and C 2?
4 is another way of showing that an argument is correct. Definitions: Literal: A variable or a negation of a variable is called a literal. Sum and Product: A disjunction of literals is called a sum and a conjunction of literals is called a product. Clause: A disjunction of literals is called a clause. Resolvent: For any two clauses C 1 and C 2, if there is a literal L 1 in C 1 that is complementary to literal L 2 in C 2, then delete L 1 and L 2 from C 1 and C 2 respectively and construct the disjunction of the remaining clauses. The constructed clause is a resolvent of C 1 and C 2. C 1 = P Q R C 2 = P S T What is a resolvent of C 1 and C 2? Q R S T
5 Theorem Given two clauses C 1 and C 2, a resolvent C of C 1 and C 2 is a logical consequence of C 1 and C 2. Example: Modus ponens (P (P Q) Q) C 1 : P C 2 : P Q The resolvent of C 1 and C 2 is Q which is a logical consequence of C 1 and C 2.
6 Theorem Given two clauses C 1 and C 2, a resolvent C of C 1 and C 2 is a logical consequence of C 1 and C 2. Definition (Resolution principle and refutation) Given a set S of clauses, a (resolution) deduction of C from S is a finite sequence C 1,..., C k of clauses such that each C i either is a clause in S or a resolvent of clauses preceding C and C k = C. A deduction of (empty clause) is called a refutation.
7 Theorem Given two clauses C 1 and C 2, a resolvent C of C 1 and C 2 is a logical consequence of C 1 and C 2. Definition (Resolution principle and refutation) Given a set S of clauses, a (resolution) deduction of C from S is a finite sequence C 1,..., C k of clauses such that each C i either is a clause in S or a resolvent of clauses preceding C and C k = C. A deduction of (empty clause) is called a refutation of a proof of S. If there is an argument where P 1,..., P r are the premises and C is the conclusion, to get a proof using resolution principle, put P 1,..., P r in clause form and add to it C in clause form. From this sequence, if can be derived, the argument is valid.
8 If there is an argument where P 1,..., P r are the premises and C is the conclusion, to get a proof using resolution principle, put P 1,..., P r in clause form and add to it C in clause form. From this sequence, if can be derived, the argument is valid. Example: What are the clauses? T (M E) S E T S M
9 Example: T (M E) S E T S M C 1 : T M E C 2 : S E C 3 : T C 4 : S C 5 : M
10 Example: T (M E) S E T S M C 1 : T M E C 2 : S E C 3 : T C 4 : S C 5 : M C 6 : T M S (resolvent of C 1 and C 2 )
11 Example: T (M E) S E T S M C 1 : T M E C 2 : S E C 3 : T C 4 : S C 5 : M C 6 : T M S (resolvent of C 1 and C 2 ) C 7 : M S (resolvent of C 3 and C 6 )
12 Example: T (M E) S E T S M C 1 : T M E C 2 : S E C 3 : T C 4 : S C 5 : M C 6 : T M S (resolvent of C 1 and C 2 ) C 7 : M S (resolvent of C 3 and C 6 ) C 8 : M (resolvent of C 4 and C 7 )
13 Example: T (M E) S E T S M C 1 : T M E C 2 : S E C 3 : T C 4 : S C 5 : M C 6 : T M S (resolvent of C 1 and C 2 ) C 7 : M S (resolvent of C 3 and C 6 ) C 8 : M (resolvent of C 4 and C 7 ) C 9 : (resolvent of C 5 and C 8 ) Hence, from the resolution principle, the argument is valid.
14 Rules of Inference for Quantified Statements
15 Rules of inference for quantified statements Rule of inference x P(x)? P(c) for an arbitrary c? x P(x)? P(c) for some element c? x(p(x) Q(x)) P(a) where a is a particular element in the domain? x(p(x) Q(x)) Q(a) where a is a particular element in the domain? Table: Rules of inference for quantified statements Name Universal instantiation Universal generalization Existential instantiation Existential generalization Universal modus ponens Universal modus tollens
16 Rules of inference for quantified statements Rule of inference x P(x) P(c) P(c) for an arbitrary c x P(x) x P(x) P(x) for some element c P(c) for some element c x P(x) x(p(x) Q(x)) P(a) where a is a particular element in the domain Q(a) x(p(x) Q(x)) Q(a) where a is a particular element in the domain P(a) Table: Rules of inference for quantified statements Name Universal instantiation Universal generalization Existential instantiation Existential generalization Universal modus ponens Universal modus tollens
17 Rules of inference for quantified statements Use rules of inference for quantified statements to show the premises A student in this class has not read the book, and Everyone in this class passed the first exam imply the conclusion Someone who has passed the first exam has not read the book.
18 End
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