induction_solutions - • (0 , 1 , 1) • if ( x,y,z ) ∈...

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SOLUTIONS TO SOME ADDITIONAL EXERCISES ON INDUCTIVE DEFINITIONS (1) (A) Give an inductive definition of the set of all finite subsets of the natural numbers. Make sure your definition is given solely in terms of the ‘ operation. (B) Give a formation sequence to show that your inductively defined set contains { 3 , 4 , 10 } . • { x } ∈ X for every natural number x . • ∅ ∈ X if A,B X , then A B X { 3 } , { 4 } , { 3 , 4 } , { 10 } , { 3 , 4 , 10 } (2) (A) Give an inductive definition of the following subset of N × N : { ( x,y ) | there exist u,v N such that 2 u = x and 3 v = y } Make sure that your definition is given solely in terms of the successor operation. (B) Show that (4 , 9) is in your inductively defined set. (0 , 0) X if ( x,y ) X , then ( s ( s ( x )) ,y ) X if ( x,y ) X , then ( x,s ( s ( s ( y )))) X (0 , 0) , (2 , 0) , (4 , 0) , (4 , 3) , (4 , 6) , (4 , 9) (3) (A) Give an inductive definition of the following subset of N × N × N : { ( x,y,z ) | x < y and x < z } Make sure your definition is given solely in terms of the successor operation. (B) Show that (2 , 4 , 3) is in your inductively defined set.
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Unformatted text preview: • (0 , 1 , 1) • if ( x,y,z ) ∈ X , then ( s ( x ) ,s ( y ) ,s ( z )) ∈ X • if ( x,y,z ) ∈ X , then ( x,s ( y ) ,s ( z )) ∈ X • if ( x,y,z ) ∈ X , then ( x,y,s ( z )) ∈ X • if ( x,y,z ) ∈ X , then ( x,s ( y ) ,z ) ∈ X (0 , 1 , 1) , (1 , 2 , 2) , (2 , 3 , 3) , (2 , 4 , 3) 1 2 SOLUTIONS TO SOME ADDITIONAL EXERCISES ON INDUCTIVE DEFINITIONS (4) (A) Give an inductive definition of the following subset of N + × N + : { ( m,n ) | n = 1 · 2 · ... · ( m-1) · m } Make sure your definition is given solely in terms of the successor operation and the multiplication operation on N . (B) Show that (3 , 6) is in your inductively defined set. 1 • (1 , 1) ∈ X • if ( m,n ) ∈ X , then ( s ( m ) ,s ( m ) · n ) ∈ X (1 , 1) , (2 , 2) , (3 , 6) 1 N + denotes the positive integers, and ‘ · ’ denotes multiplication....
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This note was uploaded on 02/27/2011 for the course PHILOSOPHY 101 taught by Professor H during the Spring '11 term at Columbia College.

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induction_solutions - • (0 , 1 , 1) • if ( x,y,z ) ∈...

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