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Homework 2 Solutions

Homework 2 Solutions - 1.4 Problem 9(a xL(x Jerry(b xyL(x...

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§ 1.4 Problem 1 ( a ): “For every number x , there is a bigger number y .” ( b ): “For any two numbers x and y , if x and y are nonnegative then so is their product.” ( c ): “For any two numbers x and y , there is another number z that is equal to their product.” § 1.4 Problem 9 ( a ): xL ( x, Jerry) ( b ): x yL ( x,y ) ( c ): x yL ( y,x ) ( d ): ¬∃ x yL ( x,y ) ( e ): x ¬ L (Lydia ,x ) ( f ): x y ¬ L ( y,x ) ( g ): x y ( x = y ) ( zL ( z,y )) ( h ): x y ( ( x negationslash = y ) ∧ ∀ z ( x = z x = y ) L (Lynn ,z ) ) ( i ): xL ( x,x ) ( j ): x y ( x negationslash = y ) → ¬ L ( x,y ) The book gives the inequivalent solution x y ( L ( x,y ) x = y ) . This says that everyone loves them- selves and nobody else, whereas my solution allows for the possibility that people might not love themselves. It is unclear from the English which is intended. § 1.4 Problem 12 ( e ): ( x ( x negationslash = Joseph) C (Sanjay ,x ) ) ( f ): x ¬ I ( x ) ( g ): ¬∀ xI ( x ) ( h ): x y ( ( x = y ) I ( y ) ) ( i ): x y ( ( x negationslash = y ) I ( y ) ) ( j ): x ( I ( x ) → ∃ y ( ( y negationslash = x ) C ( x,y ) )) ( k ): x ( I ( x ) ∧ ∀ y ( ( y negationslash = x ) → ¬ C ( x,y ) )) ( l ): x y ( ( x negationslash = y ) ∧ ¬ C ( x,y ) ) ( m ): x yC ( x,y ) ( n ): x y z ( ( x negationslash = y ) ∧ ¬ C ( x,z ) ∧ ¬ ( y,z ) ) ( o ): x y bracketleftbig ( x negationslash = y ) ∧ ∀ z ( C ( x,z ) C ( y,z ) )bracketrightbig Some of these are open to interpretation, due to the vagueness of the English language. For instance, take part ( n ), for which the solutions manual gives x y ( x negationslash = y ∧ ∀ z ¬ ( C ( x,z ) C ( y,z ) )) My solution says, “There are two people x and y and a person z with whom x and y have not chatted,” so that x and y have not chatted with the same person. The solutions manual’s solution says, “There are two
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