# asg4-sol - MATH1090 Problem Set 4-Solutions December 2007...

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MATH1090 Problem Set 4 —Solutions December 2007 Dept. of MATH and STATS MATH1090. Problem Set 4 —Solutions Posted: Dec. 5, 2007 Do the following problems from the text — each has max 5 MARKS . (1) Section 6.6: Problems: 2: Show that ( x )( A ( B C )) ± ( x )( A B ) ( x )( A C ) ² . Proof. Instead do: ( x )( A ( B C )) ( x )( A B ) ( x )( A C ). Now use ping-pong: ( ) This is ( x )( A ( B C )) , ( x )( A B ) ( x )( A C ). (1) ( x )( A ( B C )) h hyp i (2) ( x )( A B ) h hyp i (3) A ( B C ) h (1) + spec i (4) A B h (2) + spec i (5) A C h (3,4) + Post i (6) ( x )( A C ) h (5) + gen; OK, lines (1,2) have no free x i ( ) This is ( x )( A ( B C )) , ( x )( A C ) ( x )( A B ) and the proof is omitted being the same with the above (interchanging B and C ). ± 3: Show that ( x )(( A B ) C ) ( x )( A C ). Proof. Instead do: ( x )(( A B ) C ) ( x )( A C ). (1) ( x )(( A B ) C ) h hyp i (2) A B C h (1) + spec i (3) A C h (2) + Post i (4) ( x )( A C ) h (3) + gen; OK, line (1) has no free x i ± 4: Show that ( x )( A ( B C )) ( x )( A B ). Page 1 G. Tourlakis

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MATH1090 Problem Set 4 —Solutions December 2007 Proof. Instead do: ( x )( A ( B C )) ( x )( A B ). (1)
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## This note was uploaded on 05/09/2010 for the course LAPS CSE 1090 taught by Professor Peter during the Winter '10 term at York University.

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asg4-sol - MATH1090 Problem Set 4-Solutions December 2007...

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