In 10—14 S denotes the set of real numbers strictly between 0 and 1. That is, .
* 14. Define a function g from the set of real numbers to S by the following formula:
For each real number x,
Prove that g is a one-to-one correspondence. (It is possible to prove this statement either with calculus or without it.) What conclusion can you draw from this fact?
To prove that is a one-to-one correspondence
One to one: Every element in domain must be mapped with element in codomain.
Let be the set of all real numbers that are between .
Define the function , for all .
The objective is to show that is a one to one correspondence.
The function is one to one.
Therefore, is one to one
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