Construct a bijection between 01 and 01 to show that these s

Construct a bijection between [0,1] and (0,1] to show that these sets are equinumerous. You do not need to formally prove that your function is both injective and surjective.

Solution

Denote first set by A and second set by B

Let, S1={0,1,1/2,1/3,1/4,..,1/n,.....} be a subset of A

S2={1,1/2,1/3,....,1/n,....} be a subset of B

LEt, h be map from S1 to S2

h(0)=1

h(1/n)=1/(n+1)

So, h is a bijection from S1 to S2

So we have the bijection from A to B

f(x)=x if x is in A-S1

else

f(x)=h(x)