Question

Find an inductive definition for S = {3, 4, 5, 8, 9, 12, 16, 17, …}

Answer 1

Set is said to be induction that shold be satisfy the following three conditons

1. Basis: a set should conatains at least one element

2. Induction: construct as possible as many rules from existing elements in the set

3.Clousure : The new elements came out from the rules those elements should be in Set S

  Find an inductive definition for S = {3, 4, 5, 8, 9, 12, 16, 17, 20, 24, 33,…}. Solution: To simplify things we might try to “divide and conquer” by writing S as the union of more familiar sets as follows: S = {3, 5, 9, 17, 33, …} ∪ {4, 8, 12, 16, 20, 24, …}. Basis: 3, 4 ∈ S. Induction: If x ∈ S then (if x is odd then 2x – 1 ∈ S else x + 4 ∈ S). Example 3: Describe the set S defined inductively as follows: Basis: 2 ∈ S; Induction: x ∈ S implies x ± 3 ∈ S.