Question

Give a recursive definition of

1. the set of even integers.
2. the set of positive integers congruent to 2 modulo 3.
3. the set of positive integers not divisible by 5.

Short answer

$\begin{array}{r}\left(\mathrm{a}\right)\in \mathrm{S}\\ \mathrm{s}+2\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\\ \mathrm{s}-2\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\end{array}$

$(\mathrm{b})2\in \mathrm{S}\mathrm{and}\mathrm{s}+3\in \mathrm{S}\mathrm{whenever}\mathrm{s}\in \mathrm{S}$

$\begin{array}{r}\left(\mathrm{c}\right)1\in \mathrm{S}\\ 2\in \mathrm{S}\\ 3\in \mathrm{S}\\ 4\in \mathrm{S}\\ \mathrm{s}+5\in \mathrm{S}\text{whenever}\mathrm{s}\in \mathrm{S}\end{array}$

Step-by-step solution

The recursive definition of the sequence:

The recursive sequence is a sequence of numbers indexed by an integer and generated by solving a recurrence equation.

(a) To give a recursive definition of the set of even integers

$\mathrm{s}\in \mathrm{S}$Let S is the set of even integers.

First add the even integer 0.

$0\in \mathrm{S}$

Now, every even integer is the previous even integer increased by 2.

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

Also, every integer is the next even integer decreased by 2.

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

(b) To give a recursive definition of the set of positive integers congruent to 2 modulo 3

Let S is the set of positive integers congruent to 2 modulo 3.

The first positive integer congruent to modulo 3 is 2.

$2\in \mathrm{S}$

Now, every positive integer congruent to modulo 3 is the previous positive congruent to is increased by 3.

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

(c) To give a recursive definition of the set of positive integers not divisible by 5.

Let S is the set of positive integers not divisible by 5.

Now add the first positive integers congruent to 1 modulo 5, 2 modulo 5, 3 modulo 5, 4 modulo 5 respectively.

$\begin{array}{r}1\in \mathrm{S}\\ 2\in \mathrm{S}\\ 3\in \mathrm{S}\\ 4\in \mathrm{S}\end{array}$

Next, every positive integer congruent to 1 modulo 5 is the previous positive congruent to 1 modulo 5 increased by 5.

Therefore, $\mathrm{s}+5\in \mathrm{S}$whenever $\mathrm{s}\in \mathrm{S}$.