2

I need some help on homework. Here is the problem I am stuck on:

Prove that every closed interval [a,b] is numerically equivalent to [0,1]

I believe that I need to find an injection between the two sets. But I'm not sure how to get there. Any help would be great! Thanks

user69839
  • 113
  • 3
  • 9

1 Answers1

2

Look for a function $f$ of the shape $f(x)=kx+l$. Your conditions will determine $k$ and $l$. It is simplest to go from $[0,1]$ to $[a,b]$. So we want $l=a$ and $k+l=b$.

André Nicolas
  • 507,029
  • Do I actually need to find a function using numbers between 0 and 1? like a decimal expansion sort of thing? The book I am learning from mentions that. – user69839 Apr 16 '13 at 02:58
  • If you are trying to prove that the two intervals have the same cardinality, then any function which is one to one and onto (any bijection) will do the job. – André Nicolas Apr 16 '13 at 03:13
  • ok I got it! F(x)=(b-a)x+a works correct? – user69839 Apr 16 '13 at 03:17
  • Good, you do have it. It is a little harder, but not much, to find the $k$ and $l$ such that the function $g(x)=kx+l$ sends $[a,b]$ to $[0,1]$. For completenness, you should show that your $f(x)$ is one to one and onto. For example, to show $f$ is one to one, you show that if $(b-a)x+a=(b-a)x'+a$, then $x=x'$. That's just some algebra. – André Nicolas Apr 16 '13 at 03:24