How to prove the open intervals (0,1) and (0,2) have the same cardinalities? |(0, 1)| = |(0, 2)|(adsbygoogle = window.adsbygoogle || []).push({});

Let a, b be real numbers, where a<b. Prove that |(0, 1)| = |(a, b)|

-----------------------------------

|(0,1)| = |R| = c by Theorem

-----------------------------------

I know that we need to construct a function f: (0,1)->(0,2) and prove f is bijection so that |(0, 1)| = |(0, 2)|

same process of proving |(0, 1)| = |(a, b)|

but how to construct a function f: (0,1)->(0,2)

and how to construct a function g: (0,1)->(a,b) where a<b and a,b are real numbers?

I know how to construct a function f: (0,1)->R

by define a function f(x)=(1-2x)/[x(x-1)] where x cannot be 0 and 1 and when the middle domain(f)=1/2, f(1/2)=0

How can I expand this knowledge and to define a function that the domain(f) is within (0,1) and the range(f) falls into (0,2) or any close interval (a,b)?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Cardinalities of Sets: Prove |(0, 1)| = |(0, 2)| and |(0, 1)| = |(a, b)|

**Physics Forums | Science Articles, Homework Help, Discussion**