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        riddles >> hard >> What  An  Odd  Set  Of  I
        
(Message started by: K Sengupta on Dec 11^th, 2005, 11:02pm)

Title: What An Odd Set Of I
Post by K Sengupta on Dec 11^th, 2005, 11:02pm

Let S be the set of all odd integers greater than one. For each x c S ( belonging to S ) let us denote by d(x) the unique integer satisfying the undernoted inequality:

2^d(x) < x < 2^{d(x) + 1}

For both A ,B c S, define

A # B = 2^{d(A) -1} *(B-3) + A ;
For example, to calculate 5 # 7, note that
2² < 5 < 2³. So, d(5) =2 giving, 5 # 7 =2^2-1* (7-3) + 5 = 13.

PROVE that if A,B,C c S;

(I) A # B c S and,
(II) (A # B) # C = A # ( B # C);

Title: Re: What An Odd Set Of &nb
Post by Icarus on Dec 12^th, 2005, 4:33pm

Pardon my edit, but I think this is easier to follow than using "E" as an element sign in the midst of other capital letters used as variables.

Title: Re: What An Odd Set Of &am
Post by Grimbal on Dec 13^th, 2005, 2:14am

Isn't ( more readable than c?

Title: Re: What An Odd Set Of I
Post by Joe Fendel on Dec 13^th, 2005, 1:02pm

It looks to me like (I) is trivial (since an even number plus an odd one is an odd number), and (II) simplifies to proving that

d(A # B) + 1 = d(A) + d(B)

for all odd A, B > 1.

Title: Re: What An Odd Set Of &am
Post by Icarus on Dec 13^th, 2005, 3:03pm

on 12/13/05 at 02:14:36, Grimbal wrote:

Isn't ( more readable than c?

I hadn't thought of that. (I really wish we could have our mathematical beastiary back! :'()