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

(In reply to Solutions 2 to 4 by Old Original Oskar!)

(2) From point (x,y) in cartesian coordinates go to point (x,2ðy) in polar notation.

Wouldn't all points (0,y) go to the origin: (0,2ðy)

(3) From point x go to point (1,2ðx) in polar notation.

This suffers the endpoint problem: (1,0) = (1,2ð)

(4) If z=0.abcdef..., set x=0.ace... and y=0.bdf..., and apply recipe (2).

Same problem as (2) all points .0b0d0f... go to origin

Posted by Jer on 2006-09-21 20:28:18