• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Proving something that is equinumerous

Problem Statement
Prove that (0,1) ≈ R.
Relevant Equations
arctan: R→(-π/2,π/2)
arctan(x)=y ⇔ tan(y)=x for x∈R and y∈(-π/2,π/2)
I am slightly confused as to how to prove this. I know that two sets are equinumerous if there is a bijection between them. So we are trying to find f: (0,1)→R? It told us that we may assume the inverse tangent function so that would mean
arctan: R→(-π/2,π/2). This satisfies arctan(x)=y ⇔ tan(y)=x for x∈R and y∈(-π/2,π/2). We were also given that we could assume arctan is a bijective without proof. I am having trouble moving forward from this information and how to construct a proof out of all of these.

Thank you.
 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,460
714
I am slightly confused as to how to prove this. I know that two sets are equinumerous if there is a bijection between them. So we are trying to find f: (0,1)→R? It told us that we may assume the inverse tangent function so that would mean
arctan: R→(-π/2,π/2). This satisfies arctan(x)=y ⇔ tan(y)=x for x∈R and y∈(-π/2,π/2). We were also given that we could assume arctan is a bijective without proof. I am having trouble moving forward from this information and how to construct a proof out of all of these.

Thank you.
Think about how to get a bijection between ##(0,1)## and ##(-\frac \pi 2,\frac \pi 2)## so you can combine it with the given arctan bijection to establish your 1-1 correspondence.
 
Think about how to get a bijection between ##(0,1)## and ##(-\frac \pi 2,\frac \pi 2)## so you can combine it with the given arctan bijection to establish your 1-1 correspondence.
I found f(x)=(1/2)+(x/π). But if we combine with arctan would it then be f(x)=(1/2)+(1/π)arctan(x)?
 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,460
714
I found f(x)=(1/2)+(x/π). But if we combine with arctan would it then be f(x)=(1/2)+(1/π)arctan(x)?
Can't you tell me instead of asking? Does that work?
 
Can't you tell me instead of asking? Does that work?
I am slightly confused if it works. When I input my values I get numbers (0,1) or 0<x<1, so yes unless I am not understanding but this is how I interpreted it.
 

LCKurtz

Science Advisor
Homework Helper
Insights Author
Gold Member
9,460
714
You have arctan which is a 1-1 function from R onto ##(-\frac \pi 2,\frac \pi 2)## and a function f which is (you need to prove) one to one from ##(-\frac \pi 2,\frac \pi 2)## onto ##(0,1)##. Then you need to show or have a theorem about the compositions of bijections being a bijection.
 

Want to reply to this thread?

"Proving something that is equinumerous" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top