- #1

kingwinner

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## Homework Statement

Q1) Assuming that |R|=|[0,1]| is true, how can we rigorously prove that |R

^{2}|=|[0,1] x [0,1]|? How to define the bijection? [Q1 is solved, please see Q2]

Q2)

**Prove that |[0,1] x [0,1]| ≤ |[0,1]|**

__Proof:__Represent points in [0,1] x [0,1] as infinite decimals

x=0.a

_{1}a

_{2}a

_{3}...

y=0.b

_{1}b

_{2}b

_{3}...

Define f(x,y)=0.a

_{1}b

_{1}a

_{2}b

_{2}a

_{3}b

_{3}...

To avoid ambiguity, for any number that has two decimal representations, choose the one with a string of 9's.

f: [0,1] x [0,1] -> [0,1] is one-to-one, but not onto.

This one-to-one map proves that |[0,1] x [0,1]| ≤ |[0,1]|.

Now

**how can we formally prove that f is a one-to-one map (i.e. f(m)=f(n) => m=n)?**All textbooks are avoiding this step, they just say it's obviously one-to-one, but this is exactly where I'm having trouble. How to prove formally?

## Homework Equations

## The Attempt at a Solution

As shown above.

Thanks a million! :)

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