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From a pdf textbook:

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

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?

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|.__Example__(inﬁnite sets having the same cardinality). Let f : (0, 1) → (1,∞) bedeﬁned 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.
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