Question

Give a recursive definition of

a) the set of odd positive integers.
b) the set of positive integer powers of 3.
c) the set of polynomials with integer coefficients.
For example: 5x 3 − 2x 2 + 3 or 7x 4 − 8x 3 + x

Answer 1

a. for positive odd integers starting 1 and increaments by 2:

( first odd number ) + 2 ∈ S

b. for positive integer powers of 3 :

3( integer ) ∈ S

c. for integer coefficient of the given polynomial :

s. t ∈ S , s - t ∈ S and s + t ∈ S

Explanation:

the odd number starts with 1 and increases by two, and the set of that would range from 1 to infinity.