Showing that two intervals have the same cardinality ?

[SMA_01]

Homework Statement

I need to show that [0,1] and [0,2] have the same cardinality by giving a formula for a function that is bijective. Aren't there a number of functions that can fit this description? Can I then use any one? I'm a little confused, my teacher didn't really elaborate much upon this. Is there a specific formula to find the function, not only in this case, but generally?


Edit: I have used y=f(x)=2x, I'm guessing this process is pretty systematic when dealing with such intervals?

 

[Curious3141]

Homework Statement

I need to show that [0,1] and [0,2] have the same cardinality by giving a formula for a function that is bijective. Aren't there a number of functions that can fit this description? Can I then use any one? I'm a little confused, my teacher didn't really elaborate much upon this. Is there a specific formula to find the function, not only in this case, but generally?Edit: I have used y=f(x)=2x, I'm guessing this process is pretty systematic when dealing with such intervals?

You can use any bijection. f(x) = 2x is fine. So is a non-linear one like f(x) = x(3-x), which is bijective over [0,1]. It's your choice. But the former has the advantage of being immediately and intuitively obvious.

 

[Tomer]

Homework Statement

Is there a specific formula to find the function, not only in this case, but generally?

There's definitely no "formula" here. Intuition and experience are your best friends when seeking for Bijections.