asg4-sol-addendum - MATH1090 Problem Set 4 -Solutions...

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MATH1090 Problem Set 4 —Solutions addendum December 2007 Dept. of MATH and STATS MATH1090. Problem Set 4 —Solutions addendum Posted: Dec. 5, 2007 (1) Section 6.6: Problems: 11: Prove ( x )( x := t A ) A [ x := t ], provided x is not free in t . Proof. ( x )( x := t A ) ⇔ h Def of ∃i ¬ ( x ) ¬ ( x := t A ) ⇔ h WL + taut. denom ¬ ( x ) p i ¬ ( x )( x := t → ¬ A ) ⇔ h WL + 1-pt rule ( version); denom ¬ p i ¬¬ A [ x := t ] ⇔ h taut. i A [ x := t ] ± 17: Prove by aux. var. metathm: ( x )( A B ) ( x ) A ( x ) B . Proof. Instead I prove ( x )( A B ) , ( x ) A ( x ) B . But this I did in class (it is in the text too). ± 20: About the “proof” of φ ( x,y ) φ ( x,y ). 1) 6.1.19 cannot be applied because it requires φ ( x,y ) in order to con- clude φ ( y,x ). But I do not have φ ( x,y ). 2) That no proof exists follows by building a countermodel:

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asg4-sol-addendum - MATH1090 Problem Set 4 -Solutions...

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