Discrete math - Infinite sets having the same cardinality.

[DeltruS]

From a pdf textbook:
Example (infinite sets having the same cardinality). Let f : (0, 1) → (1,∞) be
defined by f(x) = 1/x. Then f is a 1-1 correspondence. (Exercise: prove it.) Therefore,
|(0, 1)| = |(1,∞)|.

Exercise. Show that |(0,∞)| = |(1,∞)| = |(0, 1)|. Use this result and the fact that
(0,∞) = (0, 1) ∪ {1} ∪ (1,∞) to show that |(0, 1)| = |R|.

This example greatly confused me. A 1-1 correspondence (aka a bijection) needs a unique value in the domain for each value in the range. There are only two values, (0,1), that map to Z⁺, so how can it be a 1-1 correspondence?

I also do not understand what the function f(x) = 1/x would look like with only (0,1) as its domain. Could someone expand what this would look like?

EDIT: solved. I understand now. (0,1) is an interval containing all decimal values between 0 and 1. I thought this question was asking for an infinite string of binary values or something like that.

 

[lippyka]

The function f(x) = 1/x would look like this: { (x,f(x)) | x ∈ (0,1) } = { (x, 1/x) | x ∈ (0,1) }. This means that for any value x in the domain (0,1), the corresponding value in the range is 1/x. For example, f(0.5) = 1/0.5 = 2. Thus, (0.5,2) is a pair in the bijection.