# hwk2 - CSE 2315 Discrete Structures Homework 2 Predicate...

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CSE 2315 - Discrete Structures Homework 2: Predicate Calculus and Proof Techniques CSE 2315 - Discrete Structures Homework 2- Fall 2010 Due Date: Oct. 7 2010, 3:30 pm Proofs using Predicate Logic For all your predicate logic proofs you can use only the rules given in the following tables. In addition you are allowed to apply the deduction method and to use the method of temporary hypotheses. All other rules have to be proven ﬁrst. Equivalence Rules Rule Name Expression Equivalent Expression Commutativity (comm) P Q Q P P Q Q P Associativity (ass) ( P Q ) R P ( Q R ) ( P Q ) R P ( Q R ) Distributivity (dis) ( P Q ) R ( P R ) ( Q R ) ( P Q ) R ( P R ) ( Q R ) De Morgan’s Laws (De Morgan) ( P Q ) P Q ( P Q ) P Q Implication (imp) P Q P Q Double negation (dn) ( P ) P Self-reference (self) P P P Negation (neg) ( x ) P ( x ) ( x ) P ( x ) 2010 Manfred Huber Page 1

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CSE 2315 - Discrete Structures Homework 2: Predicate Calculus and Proof Techniques Inference Rules Rule Name From Can Derive Conjunction (con) P , Q P Q Simpliﬁcation (sim) P Q P , Q Modus ponens (mp) P , P Q Q Modus tollens (mt) P Q , Q P Addition (add) P P Q Universal instantiation (ui) ( x ) P ( x ) P ( y ) ( Be careful with the rule’s restrictions) ( x ) P ( x ) P ( a ) Existential Instantiation (ei) ( x ) P ( x ) P ( y ) ( Be careful with the rule’s restrictions) ( x ) P ( x ) P ( a ) Universal generalization (ug) P ( x ) ( x ) P ( x ) ( Be careful with the rule’s restrictions) Existential generalization (eg) P ( x ) ( x ) P ( x ) ( Be careful with the rule’s restrictions) P ( a ) ( x ) P ( x ) For all proofs the steps have to be annotated such as to indicate the rule and which elements of the proof sequence it was applied to. 1. For each instantiation and generalization step in the following proof sequences indicate if it is legal. You have to justify your decision (a short justiﬁcation is sufﬁcient). a)
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hwk2 - CSE 2315 Discrete Structures Homework 2 Predicate...

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