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In each of (a) to (d) you have to find a strictly increasing surjective function from A to B or prove that there doesn't exist any. (a)

A=\{x|x\in \mathbb{Q},x\leq \sqrt{2}\}

and

B=\{x|x\in \mathbb{Q},x\leq \sqrt{3}\}

(b)

A=\mathbb{Q}

and

B=\mathbb{Q}\cup \{\pi \}

In (c) and (d) we say

(x,y)>(z,t)

where

x,y,z,t \in \mathbb{R}

, whenever

x>z

x=z

and

y>t

. (c)

A=\mathbb{R}

and

B=\mathbb{R}^2

(d)

X=\{2^{-x}| x\in \mathbb{N}\}

, then

A=X \times (X\cup \{0\})

and

B=(X \cup \{ 0 \}) \times X

(e) If

A,B \subset \mathbb{R}

, such that there exists a surjective non-decreasing function from

A

B

and a surjective non-decreasing function from

B

A

, does there exist a surjective strictly increasing function from

B

A

?
Time allowed for this problem was 2 hours.

Problem Statement