Recursive Properties As They Relate To Domains I'm struggling with understanding what restrictions recursive properties generally place on domains. also on a more global level I don't think I fully understand how functions are described in symbolic logic.
I hope these two examples will illustrate these issues:


*

*Say there's a function $g:N \rightarrow N$ that satisfies the
recursive property $g(n+1) = 7n + 2g(n)$ $\forall n \in N$
My first question is this: Let's say I define a new function $g(n) =
    n^2 + 5n + 2$. What's the difference between $g(n)$ and the $g$ I
described earlier? If I went on and described another $g(n)$ as
being $g(n) = 10n$, what claim would I be making about both $g(n) =
    n^2 + 5n + 2$ and $g(n) = 10n$? Is this allowed? Are they the same?
Are they different?

*Finally, if I stated that $g(n) = 10n $ holds true for $\forall n
    \in N$, would I be making the following claim?
$\forall n \in N, g(n) = 10(n) \Rightarrow g(n+1) = 7n + 2g(n)$ 
or would I be making the following claim?
$\forall n \in N, g(n) = 10(n) \Leftrightarrow g(n+1) = 7n + 2g(n)$
 A: Ok, firstly, when you want to define a new function PLEASE use a new symbol, especially when you go on to reference both.
So I am going to say $f(n+1)=7n+2f(n), g(n)=n^2+5n+2, h(n)=10n$.
Now $$g(n+1)=(n+1)^2+5(n+1)+2=n^2+2n+1+5n+5+2=n^2+5n+2+2n+6=g(n)+2n+6$$ 
So firstly $g$ does not satisfy the recurrence relation for $f$. Also notice at this point that we know precisely what function $g$ is, i.e. on input $n$ we can work out $g(n)$. However we don't know how to calculate $f$, we only know how to calculate it if we know smaller values, which we don't. So the recursive relation which $f$ satisfies does not uniquely specify the function $f$, we need to specify $f(0)$.
To illustrate if we have a function $i(n)$ which satisfies $i(n)=i(n)+2n+6$ do we know that $i=g$?
I don't really know what you are getting at regarding $h$, certainly $g$ and $h$ are different functions, for instance $g(0)=2$ and $h(0)=0$.
For two you are getting really confused, sentences are false as $g(0)=10(0)=0$ yet $g(1)\neq 7 + 0$. 
