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s09_t05_06 - UNIVERSITY OF WATERLOO School of Computer...

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UNIVERSITY OF WATERLOO School of Computer Science CS245 Logic and Computation Spring 2009 TUTORIALS 5 and 6 1. Show that x p ( x ) m ( x ), x h ( x ) p ( x ) 6| = x m ( x ) h ( x ). In other words, show that x m ( x ) h ( x ) does not logically follow from the premises. To show that the conclusion does not logically follow from the premises, we must find an interpretation in which the premises are true but the conclusion is false. I.e., we must find a counter example. It is a good idea when looking for a counter example to start with small domains of discourse. So, let the domain of discourse D = { A } . Consider the following interpretation: m (A) is true h (A) is false p (A) is true We can verify that the premises are true in this interpretation but the conclusion is false. 2. Transformational proof of ¬ ( x p ( x ) ∨ ∃ x q ( x )) WV x • ¬ p ( x ) ∧ ¬ q ( x ) ¬ ( x p ( x ) ∨ ∃ x q ( x )) WV ¬ ( x p ( x ) q ( x )) -Distrib- WV x • ¬ ( p ( x ) q ( x )) De Morgan WV x • ¬ p ( x ) ∧ ¬ q ( x ) De Morgan 1
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3. Natural Deduction for Predicate Logic The next two examples involve redoing examples used in the Nissanke text in the style of sub-proofs and without the logical laws of transformational proof. They are quite challenging.
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