Answer to Question #136797 in Discrete Mathematics for ashmita

Question

Prove or disprove that there exists a bijection from (0, 1] to [0, ∞)^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, ∞)²
2. Slightly modify the above function to prove that (0,1]  is equivalent to [0, ∞)²
3. Prove that (0,1]  is equivalent to [0, ∞)²

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