I need to show that is a bijection of on
I think I need to show that the expression is both injective and surjective, but I am not sure how to do that.
Maybe a kind person can help me in some direction?
Thanks in advance.
Thanks for your reply. I am not quite sure how to show that every natural number is an odd number (possibly 1) times some non-negative integer power of 2. I can see it works by doing some examples, but how I precisely can show it? Hmm, maybe a hint?
Definition of "even": a natural number, n is even if and only if n= 2k where k is a natural number.
If N is a natural number, it can be written as N= 2K for some natural number K. If K is odd, you are done- N is "2 to the 1 power" times n odd number. if K is even, K= 2M for some integer M. If M is odd, you are done: N= 2K= 2(2M)= 4M is "2 to the 2 power" times an odd number. If M is even, M= 2J for some integer J. If J is odd....
Note that N> K> M> J... so that will eventually terminate.