Let A be a set such $$A=(1,2]\cup{((3,4)\cap{\mathbb{Q}})}$$ and let B be a set such $$B=(0,1]\cup{(2,3]}\cup{((3,5)\cap{\mathbb{Q}})}$$

I need to find if B is isomorphic to A. I know that a bijection from A to B is needed, but I can't find a proper bijection.

It is easy to find a bijection between $(1,2]$ and $(0,1]$ - $f(x)=x-1$, but I can't define a bijection from $(3,4)$ to $(2,5)$.

How may I continue? is it even good start defining $f(x)=x-1$ for $x\in{(1,2]}$?

Please help, thank you!


After reading the comments, I realized that I need to construct 3 bijections:

  1. $f: \ (3,4)\cap{\mathbb{Q}} \rightarrow (3,5)\cap{\mathbb{Q}}$ - I constructed $f(x)=2x-3 \ , \ x\in{\mathbb{Q}}$.
  2. $g: \ \left(1,1 \frac{1}{2} \right] \rightarrow \left(0,1 \right]$ - I constructed $g(x)=2(x-1) \ , \ x\in{\mathbb{R}}$.
  3. $h: \ \left(1 \frac{1}{2},2 \right] \rightarrow \left(2,3 \right]$ - I constructed $h(x)=2\left(x-\frac{1}{2} \right)$

Finally, I defined $\phi(x)$ to be the proper function (f,g,h) in each range of x.

If there any corrections needed, please show them. Thanks again!

  • $\begingroup$ I don't think defining $f(x) = x-1$ will help. That covers all of $(0,1]$, but leaves you with nothing left to cover $(2,3]$. $\endgroup$ – MJD Dec 30 '13 at 17:17


To map $(3,4)\cap\Bbb Q$ to $(3,5)\cap\Bbb Q$ try to construct a linear map from $(3,4)$ to $(3,5)$. A linear map will take rationals to rationals in both directions.

Then you need to map $(1,2]$ to $(0,1]\cup (2,3]$. Try breaking $(1,2]$ into two disjoint parts. Map one part to $(0,1]$ and the other part to $(2,3]$.

  • $\begingroup$ Great answer :) tnx. $\endgroup$ – Galc127 Dec 30 '13 at 17:24

$$\text{Construct the following bijections and put them together:}$$

$$\text{$(3,4) \cap \mathbb{Q}$ and $(3,5) \cap \mathbb{Q}$}$$ $$\text{$(0,1]$ and $(1,1.5]$}$$ $$\text{$(2,3]$ and $(1.5,2]$}$$

  • $\begingroup$ Great answer :) tnx. $\endgroup$ – Galc127 Dec 30 '13 at 17:25

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