How Do You Construct Bijections Between Different Types of Intervals?

[panderse]

Homework Statement

Let w,x,y,z be real #'s with w<x and y<z

Construct bijections
[w,x] <-> [y,z]
(w,x] <-> [y,z)

Homework Equations

The Attempt at a Solution

So for the closed interval bijection, I was trying to work with the following:

(z-y)/(x-w) * (f-w) + y where w,x,y,z are the #'s and f is the function variable.

If I'm not mistaken, this is injective and surjective, thus bijective.

One thing I was wondering is simply what it means to "construct" the bijection?? Does it just mean presenting the equation above and showing that it is 1-1 and onto? Or is there something more complex at work?

And for the (...] <-> [...) portion, I am at a loss. Can this be done in a regular function type format, or do I need to do it piecewise?

Thanks
Pete

 

[tiny-tim]

Welcome to PF!

Hi Pete! Welcome to PF!

One thing I was wondering is simply what it means to "construct" the bijection?? Does it just mean presenting the equation above and showing that it is 1-1 and onto? Or is there something more complex at work?

Yes, "construct" just means presenting an equation.

And for the (...] <-> [...) portion, I am at a loss. Can this be done in a regular function type format, or do I need to do it piecewise?

oh come on …

imagine you're a three-year-old child and you're presented with two [) and (] -shaped bricks