
setup an induction
Hi,
Could someone help me with setting up a problem . So i want to prove something by induction on k = 0,1,2,3,4...n, clearly $\displaystyle k \in Z+ $
I have two functions: g(x) and f(x) and i have observed that
$\displaystyle g(0) = y $
$\displaystyle f(0) = g(0) = y $
$\displaystyle f(1) = g(f(0))+g(g(f(0))) $
$\displaystyle f(2) = g(f(1))+g(g(f(1)))$
$\displaystyle ...$
so it looks like :
$\displaystyle f(k) = g(f(k1))+g(g(f(k1)))$ and if i check that for all k up to 100 i get correct numbers.
And as I said i would like to prove this ($\displaystyle f(k) = g(f(k1))+g(g(f(k1)))$) by induction on k So my question is what is my base case and what my induction hypothesis? To me it looks like my base case is :
$\displaystyle g(0) = y $
$\displaystyle f(0) = g(0) = y $
and my induction H:
$\displaystyle f(k) = g(f(k1))+g(g(f(k1)))$
but what confuses me is if if i plug in 0 for k here i get :
$\displaystyle f(0) = g(f(0))+g(g(f(0)))$
$\displaystyle f(0) = g(y)+g(g(y))$
which is not y which means that either i have wrong idea about my base case (although i always recieve f(0) = g(0)) or my induction hypothesis is wrong (but for all k > 0 i always get the right numbers if i apply $\displaystyle f(k) = g(f(k1))+g(g(f(k1)))$ ). I don't know how to connect these two together.
One more information . This ofcourse works if y = 0 but y = 1,2,3..n so $\displaystyle y\in N$ and $\displaystyle g(y)> y$
Thank you
baxy

Re: setup an induction
Hi,
You made a small error in plugging in 0 for k.
If you were to plug in 0, you would get
$\displaystyle f(0) = g(f(1)) + g(g(f(1)))$
Now, $\displaystyle f$ probably is not defined on 1, so what you are trying to prove is this:
1. For $\displaystyle n=0$, $\displaystyle f(0) = g(0) = y$.
2. For $\displaystyle n=1, 2, 3, ...$, $\displaystyle f(k) = g(f(k1)) + g(g(f(k1)))$.
The second part can be proved by induction, proving the following two subparts seperately:
2a. $\displaystyle f(1) = g(f(0)) + g(g(f(0))$
2b. If $\displaystyle f(k) = g(f(k1))) + g(g(f(k1)))$ (induction hypothesis), then $\displaystyle f(k+1) = g(f(k))) + g(g(f(k)))$
Hope this helps :)