Let S be the subset of the set of ordered pairs of integers defined recursively by:

Basis step: (0,0)S.

Recursive step: If (a,b)S, then (a+2,b+3)S and (a+3,b+2)S.

a. List the elements of S produced by the first five applications of the recursive definition.

b. Use strong induction on the number of applications of the recursive step of the definition to show that 5|a+b when (a,b)S.

c. Use structural induction to show that 5|a+b when (a,b)S.