# induction - N × N × N x,y,z | x< y and x< z Make sure...

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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 } . (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. (3) (A) Give an inductive definition of the following subset of
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Unformatted text preview: N × N × N : { ( x,y,z ) | x < y and x < z } Make sure your deﬁnition is given solely in terms of the suc-cessor operation. (B) Show that (2 , 4 , 3) is in your inductively deﬁned set. (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 succes-sor operation and the multiplication operation on N . (B) Show that (3 , 6) is in your inductively deﬁned set. 1...
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