Question

Give a recursive definition of

1. The set of odd positive integers
2. The set of positive integer powers of 3
3. The set of polynomials with integer coefficients.

Short answer

$\begin{array}{r}\left(\mathrm{a}\right)1\in \mathrm{S}\text{and}\mathrm{s}+2\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\\ \left(\mathrm{b}\right)3\in \mathrm{S}\text{and}3\mathrm{s}\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\\ \left(\mathrm{c}\right)1\in \mathrm{S}\\ \mathrm{x}\in \mathrm{S}\\ \mathrm{s}+\mathrm{t}\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\text{and}\mathrm{t}\in \mathrm{S}\\ \mathrm{s}-\mathrm{t}\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\text{and}\mathrm{t}\in \mathrm{S}\\ \mathrm{s}\cdot \mathrm{t}\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\text{and}\mathrm{t}\in \mathrm{S}\end{array}$

Step-by-step solution

The recursive definition of the sequence:

A sequence can also be defined recursively, meaning that the previous terms define successive terms in the sequence. The recursive sequence is obtained by the deriving each successive term such that it is 2 larger than the previous obtained term.

Give a recursive definition of the set of odd positive integers(a)

Consider S is the set of odd positive integers such that the first odd integer is 1. Then:

$1\in \mathrm{S}$

Consider for every odd integer it is the previous odd integer increased by 2.

Therefore, $\mathrm{s}+2\in \mathrm{S}$whenever $\mathrm{s}\in \mathrm{S}$

Give a recursive definition of the set of positive integer powers of 3.(b)

Consider the positive odd integer is S.

Consider the first positive integer powers of 3 is $3^1=3$
.

Then:

$3\in \mathrm{S}$

Consider for every positive integer powers of 3 is the previous integer’s power of 3 multiplied by 3.

Therefore, $3s\in \mathrm{S}$for $s\in \mathrm{S}$.

Give a recursive definition of the set of polynomials with integer coefficients.(c)

Consider S is the set of polynomials with integer coefficients and Z is set of integers.

Now add constant 1 and then the linear term x to the set S.

$$\begin{gathered} 1\in \mathrm{S} \\ \mathrm{x}\in \mathrm{S} \end{gathered}$$

Consider the sum of two polynomials is also a polynomial including all the positive coefficients as follows:

$\mathrm{s}+\mathrm{t}\in \mathrm{S}\mathrm{for}\mathrm{s}\in \mathrm{S}\mathrm{and}\mathrm{t}\in \mathrm{S}$

Consider for the difference of two polynomials that is also a polynomial including all the negative and zero coefficients is as follows:

$\mathrm{s}-\mathrm{t}\in \mathrm{S}\mathrm{for}\mathrm{s}\in \mathrm{S}\mathrm{and}\mathrm{t}\in \mathrm{S}$

Consider the product of two polynomials is also a polynomial for $\mathrm{s}.\mathrm{t}\in \mathrm{S}$such that $\mathrm{s}\in \mathrm{S}\mathrm{and}\mathrm{t}\in \mathrm{S}$.