Question

Explain how (0,1) and R have the same cardinality. [Hint: Use the Schröder-Bernstein theorem.]

A) Both are uncountable sets
B) One-to-one correspondences exist between them
C) Schröder-Bernstein theorem doesn't apply
D) Cantor's diagonal argument explains it

Answer 1

The Schröder-Bernstein theorem states that if there exists an injective function from set A to set B and an injective function from set B to set A, then there exists a one-to-one correspondence (bijection) between the two sets. By using this theorem, we can establish that (0,1) and R have the same cardinality.

Step-by-step explanation:

The Schröder-Bernstein theorem states that if there exists an injective function from set A to set B and an injective function from set B to set A, then there exists a one-to-one correspondence (bijection) between the two sets. In the case of (0,1) and the set of real numbers (R), we can use the Schröder-Bernstein theorem to show that they have the same cardinality.

To prove this, we can establish an injective function from (0,1) to R by using a function such as f(x) = tan(π(x-1/2)) + x. This function maps every number in (0,1) to a unique real number.

Similarly, we can establish an injective function from R to (0,1) by using a function such as g(x) = 1/(1+e^x). This function maps every real number to a unique number in (0,1).

Since there exists an injective function from (0,1) to R and an injective function from R to (0,1), the Schröder-Bernstein theorem guarantees the existence of a one-to-one correspondence between the two sets. Therefore, we can conclude that (0,1) and R have the same cardinality.