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
(2) From point (x,y) in cartesian coordinates go to point (x,2πy) in polar notation.
(3) From point x go to point (1,2πx) in polar notation.
(4) If z=0.abcdef..., set x=0.ace... and y=0.bdf..., and apply recipe (2).