induction_solutions

# 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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