Answer to Question #136796 in Discrete Mathematics for ashmita

Question

Prove or disprove that there exists a bijection from (0, 1] to (0, 1]^2

Answer 1

Define f:(0,1)→R as follows.

For n∈N, n≥2, f(1/n)=1/(n−1) and for all other x∈(0,1),  f(x)=x

1.  Prove that f is a 1−1 function from (0,1) onto (0,1]²
2. Slightly modify the above function to prove that [0,1) is equivalent to [0,1]²
3. Prove that [0,1) is equivalent to (0,1]²

Since the "equivalent to" relation is both symmetric and transitive, it should follow that [0,1] is equivalent to (0,1)². Hence, there does exist a one-to-one correspondence between [0,1] and (0,1)²