# Thread: Prove that two sets have the same cardinality

1. ## Prove that two sets have the same cardinality

Hi there,

I have a question to solve: if a<b and c<d for , where a,b,c and d are all rationals, how do I prove that [a,b] and [c,d] have the same cardinality.

I know that we're supposed to show bijection between the sets, but I don't know how I would start. Any help please?

2. Can you show that both sets have the same cardinality as [0,1]?

3. See what the function $f(x)=\frac{d-c}{b-a}(x-a)+c$ does for you.

4. I got it, I think. I used Plato's function and it did the trick. Defunkt, your suggestion works too, but it might double the work if I used my current approach.

Plato, question for you: how did you get that function? i

5. Originally Posted by dgmath
Plato, question for you: how did you get that function? i
Fifty years of experience.
Moreover, any linear function is a bijection on the reals.

6. Plato, let me rephrase the question. What does that function signify? Is it some sort of a relation between the two sets [a,b] and [c,d] ?

7. He answered that: "any linear function is a bijection on the reals."

And, of course, he chose his linear function so that a is mapped to c and b is mapped to d.