303a6solns - ASSIGNMENT 6 SOLUTIONS MATH 303 FALL 2011 If...

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Unformatted text preview: ASSIGNMENT 6 SOLUTIONS MATH 303, FALL 2011 If you find any errors please let me know Manipulation (M1) By rule C we have that (( c = d ) ∧ ( d = c )) β†’ c = c is valid. Then by rule F applied with A ( x ) being (( c = x ) ∧ ( x- c )) and B being c = c we get that βˆƒ x (( c = x ) ∧ ( x = c )) β†’ c = c is valid, which is what we were supposed to show. (M2) First notice that by Cohen’s definition of β€œderive” the question is asking if ( βˆ€ xA ( x ) ∧ ( A ( c ) β†’ B )) β†’ B is valid. By rule E we know that βˆ€ xA ( x ) β†’ A ( c ) is valid. Consider the following formula ( βˆ€ xA ( x ) β†’ A ( c )) β†’ (( βˆ€ xA ( x ) ∧ ( A ( c ) β†’ B )) β†’ B ) Letting C be βˆ€ xA ( x ) this is the formula ( C β†’ A ( c )) β†’ (( C ∧ ( A ( c ) β†’ B )) β†’ B ) This is a propositional function in the letters C , A ( c ), and B , so we can apply rule A. To do this build a truth table (unfortunately a bit of a large one) A ( c ) B C C β†’ A ( c ) A ( c ) β†’ B C ∧ ( A ( c ) β†’ B ) ( C ∧ ( A ( c ) β†’ B )) β†’ B whole thing F F F T T F T T F F T F T T F T F T F T T F T T F T T F T T T T T F F T F F T T T F T T F F T T T T F T T F T T T T T T T T T T We see that the propositional function in question is identically true, so by rule A it...
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303a6solns - ASSIGNMENT 6 SOLUTIONS MATH 303 FALL 2011 If...

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