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# hwk2_sol - CSE 2315 Discrete Structures Homework 2...

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CSE 2315 - Discrete Structures Homework 2: Predicate Calculus, Proof Techniques, and Recursion CSE 2315 - Discrete Structures Homework 2- Solutions - Fall 2010 Due Date: Oct. 7 2010, 3:30 pm 1. a) 1 . ( x ) R ( x ) hyp 2 . ( x )( y )( R ( y ) P ( x,y,z )) hyp 3 . ( y )( R ( y ) P ( z,y,z )) 2 ui legal. z is not within the scope of a quantiﬁer for z . 4 . R ( z ) P ( z,a,z ) 3 ui illegal. all bound occurrences have to be replaced with the same variable or constant. 5 . R ( z ) 1 ui legal. z is not within the scope of a quantiﬁer for z . b) 1 . ( z ) Q ( x,z ) hyp 2 . ( x )( y )( Q ( x,y ) P ( x )) hyp 3 . ( y )( Q ( x,y ) P ( x )) 2 ui legal. x is not within the scope of a quantiﬁer for x . 4 . Q ( x,z ) 1 ei illegal. z occurs previously in the proof sequence. - However, it causes no error here since its only use was by the existential quantiﬁer that is removed. 5 . Q ( x,z ) P ( x ) 3 ui legal. z is not within the scope of a quantiﬁer for z . c) 1 . ( x )( y ) P ( x,y ) hyp 2 . ( x )( P ( x,a ) S ( a )) hyp 3 . ( y ) P ( x,y ) 1 ui legal. x is not within the scope of a quantiﬁer for x . 4 . P ( x,a ) 3 ei illegal. a occurs previously in the proof sequence. 5 . P ( x,a ) S ( a )) 2 ui legal. x is not within the scope of a quantiﬁer for x . 6 . S ( a ) 4 , 5 mp d) 1 . ( x ) P ( x ) hyp 2 . ( y )( P ( y ) Q ( y,z )) hyp 3 . P ( z ) 1 ei illegal. z occurs previously in the proof sequence. 4 . P ( z ) Q ( z,z ) 2 ui legal. z is not within the scope of a quantiﬁer for z . 5 . Q ( z,z ) 4 , 3 mp 6 . ( z ) Q ( z,z ) 5 eg legal. always legal for same variable name. e) 1 . ( x )( R ( x ) ( y ) Q ( x,y )) hyp 2 . ( z )( y ) Q ( z,y ) hyp 3 . R ( z ) ( y ) Q ( z,y ) 1 ui legal. z is not within the scope of a quantiﬁer for z . 4 . ( y ) Q ( z,y ) 2 ui legal. z is not within the scope of a quantiﬁer for z . 5 . R ( z ) 4 , 3 mt 6 . ( x ) R ( x ) 4 eg legal. x does not occur in 5 . 2010 Manfred Huber Page 1

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CSE 2315 - Discrete Structures Homework 2: Predicate Calculus, Proof Techniques, and Recursion f) 1 . ( x ) P ( x ) hyp 2 . ( x )(( y ) P ( y ) Q ( x,y )) hyp 3 . P ( y ) 1 ei illegal. y occurs previously in the proof sequence. 4 . ( y ) P ( y ) Q ( x,y ) 2 ui legal. x is not within the scope of a quantiﬁer for x . 5 . ( y ) P ( y ) 3 eg legal. Always legal for same variable name. 6 . Q ( x,y ) 5 , 4 mp 7 . ( x ) Q ( x,y ) 6 ug legal. x in 6 was not derived using a wff that contained x as a free variable or derived it using ei. 8 . ( y )( x ) Q ( x,y ) 7 ug illegal. y in 7 was derived from a free variable (step 2 ). g)
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hwk2_sol - CSE 2315 Discrete Structures Homework 2...

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