A function f:A→B from set A to set B is called a bijection if it is a one-to-one correspondence between A and B, i.e. for every b in B there is exactly one a in A such that f(a)=b. More informally, you could say that every element in A gets matched up with exactly one element in B and vice versa.

Can you give examples for bijections between the following sets?

1. A=(0,1), B=R

2. A=[0,1]², B=the unit disc with boundary, i.e. all points in the plane with distance smaller or equal 1 from origin

3. A=[0,1], B=the unit circle, i.e. all points in the plane with distance 1 from the origin.

4. A=[0,1], B=the unit disc with boundary

I'm no mathematician, but I think 3 is

b = f(a) = {x=sin(a*pi), y=cos(a*pi)}

(Appologies for the notation.)

One problem with this though is that f(0) and f(1) have the same values for b, so it's not a bijection. However, it works if A=[0,1) rather than A=[0,1]

Posted by bumble on 2006-09-21 15:12:16