Question

Give a recursive defintition for the following set or ordered pairs of positive integers (a|b means that a is a factor of b):
`S =`{
`(a,b)|a`∈
`Z^+, b`∈
`Z^+, a + b` is odd}

Answer 1

`(2,1)`,
`(1,2)` is your 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.

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

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

`2+6 = 8 \ \text{(even)}`

`10+12 = 22 \ \text{(even)}`

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

`3+7 = 10 \ \text{(even)}`

`5+11 = 16 \ \text{(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 your 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.