Final answer:
To show that the interval (0,1) and the set of all real numbers have the same cardinality, we employ the Schröder-Bernstein Theorem using injective functions such as the square and tangent functions that map the interval to itself and to the real numbers respectively.
Step-by-step explanation:
To demonstrate that the interval (0,1) and the set of all real numbers (denoted by ℝ) have the same cardinality, we use the Schröder-Bernstein Theorem. This theorem states that if one can find injective (one-to-one) functions from each set into the other, then the two sets have the same cardinality, meaning they have the same number of elements. To apply this to our situation, consider the square function f(x) = x² which maps the interval (0,1) to itself, and the tangent function g(x) = tan(πx/2) which maps the interval (0,1) to ℝ. Both of these are injective functions.
The inverse functions, which are also injective, demonstrate that elements from ℝ can be mapped back into the interval (0,1). These injective mappings prove that the cardinality of (0,1) is no greater than that of ℝ, and that the cardinality of ℝ is no less than that of (0,1), satisfying the criteria of the Schröder-Bernstein Theorem and confirming that their cardinalities are indeed the same.