Give a recursive definition of each of these sets of ordered pairs of positive integers. (Hint: plot the points in the set in the plane and look for lines containing points in the set.

A recursive definition, also known as an inductive definition, is used in mathematics and computer science to define the elements of a set in terms of other elements in the set.
Factorials, natural numbers, Fibonacci numbers, and the Cantor ternary set are examples of recursively-definable objects.
A recursive definition of a function defines the function's values for some inputs in terms of the function's values for other (usually smaller) inputs.
The rules, for example, define the factorial function n!

0! = 1.
(n + 1)! = (n + 1)·n!.

To give a recursive definition:

Think about how to solve this problem in general. How can we assure that the sum a+b is odd?

Think about this, what happens when we sum two even numbers? The result is even or odd?

So,

2+6 = 8 (even)
10+12 = 22 (even)

And what happens when we sum two odd numbers? The result will be even or odd?

So,

3+7 = 10 (even)
5+11 = 16 (even)

Therefore, to assure that a+b is odd, one of them has to be odd and one of them has to be even, that is why (2,1), (1,2) is the base step:

if (a,b) is in the set (a+1,b+1) will be in the set
if (a,b) is in the set (a+2,b) will be in the set
if (a,b) is in the set (a,b+2) will be in the set.

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The complete question is given below:
Give a recursive definition of each of these sets of ordered pairs of positive integers. [Hint: Plot the points in the set in the plane and look for lines containing points in the set.] a) S = {(a, b) | a ∈ Z+ , b ∈ Z+ , and a + b is odd}