Question

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.

1. huyenthanh
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.

### What is the recursive definition?

• 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!
1. 0! = 1.
2. (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.
Know more about the recursive definition here:
#SPJ4
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}