wu :: forums - Print Page


    
      
        wu :: forums
        (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi)
      

        riddles >> putnam exam (pure math) >> Intersecting Images
        
(Message started by: THUDandBLUNDER on Mar 5^th, 2005, 10:43am)

Title: Intersecting Images
Post by THUDandBLUNDER on Mar 5^th, 2005, 10:43am

Find a function R -> R such that the image of any interval intersects with every other interval.

Can such a function have an inverse?

Title: Re: Intersecting Images
Post by Obob on Mar 5^th, 2005, 1:58pm

Use the Axiom of Choice to find a set of coset representatives for the abelain group R/Q. This group is uncountably infinite (since Q is countable and R is uncountable), so there exists a bijection between the set of coset representatives and R, call this function g. Define a function f:R->R by declaring f(x)=g(x+Q); that is, f evaluated at x is equal to g evaluated at the unique coset representative which differes from x only by an element of Q. Then f is a well-defined function, and it has the property that the image of any interval is all of R since every interval contains a complete set of coset representatives for R/Q!

I haven't been able to figure out the inverse question, I'll leave that one for now.

Title: Re: Intersecting Images
Post by THUDandBLUNDER on Mar 12^th, 2005, 11:23pm

Any ideas on a function constructibe in ZF?
Is it bijective?

Title: Re: Intersecting Images
Post by Obob on Mar 13^th, 2005, 11:17am

This function is obviously not bijective. While its certainly possible that there is some such function which is constructible in ZF, it seems like it would be extremely difficult to construct.