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Unformatted text preview: a Øh4Ô Ö iÒg6ÓÑ eÛ Ó Ök# Ó ÐÙ Ø iÓÒ ×bÝ iÑC a ÐÐahaÒ § 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,Some of these are open to interpretation, due to the vagueness of the English language....
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This note was uploaded on 07/12/2008 for the course MAT 243 taught by Professor Callahan during the Spring '06 term at ASU.
 Spring '06
 CALLAHAN

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