L4[1] - Example Show that x( P( x) Q( x) and are logically...

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Example Show that and are logically equivalent. ( ( ) ( )) x P x Q x   ( ( ) ( )) ( ( ) ( )) (De Morgan) x P x Q x x P x Q x    ( ( ) ( )) (convert) x P x Q x      ( ( ) ( )) x P x Q x  ( ( ( )) ( ( ))) x P x Q x       ( ( ) ( )) (double negation) x P x Q x  
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Rules of Inference: Terms An argument is a sequence of statements that end with a conclusion. An argument is valid if the conclusion follows from the truth of the preceding statements. That is, an argument is valid if and only if it is impossible for the preceding statements (premises) to be true while the conclusion is false. A fallacy is a common form of incorrect reasoning, which lead to invalid arguments.
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Rules of Inference I
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Rules of Inference II
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Building an Argument Show that the following hypotheses… “It is not sunny this afternoon and it is colder than yesterday.” “We will go swimming only if it is sunny.” “If we do not go swimming then we will take a canoe trip.” “If we take a canoe trip then we will be home by sunset.” lead to the conclusion “We will be home by sunset.”
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Building an Argument (cont‟d) Let p be “It is sunny this afternoon.” Let q be “It is colder than yesterday.” Let r be “We will go swimming.” Let s be “We will take a canoe trip.” Let t be “We will be home by sunset.” Then the hypotheses we have to work with are The conclusion is t . , , , p q r p r s s t      
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Building an Argument (cont‟d) Again, the hypotheses are (1) (2) (3) (4) (5) (6) (7) (8) , , , p q r p r s s t     (Hypothesis) pq  (Simplification) p rp (Modus tollens using (2) and (3)) r rs  (Modus ponens using (4) and (5)) s st (Modus ponens using (6) and (7)) t
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Another Example Show that the following hypotheses… “If you send me an email message, then I will finish writing the program.” “If you do not send me an email message, then I will go to sleep early.”
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This note was uploaded on 06/01/2010 for the course MATH 1P66 taught by Professor Jeffharoutunian during the Spring '10 term at Brock University.

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L4[1] - Example Show that x( P( x) Q( x) and are logically...

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