Show equinumerosity between two sets by finding bijective functions

Question

I need to find a bijective functions to show equinumerosity between two sets and I hit problems with certain exercices:

1. $A=\mathbb{R}$ and $B= \left\{ \langle x,y \rangle |x\in \mathbb{R} \wedge y=\cos{x} \right\}$

I think the function is: $f(x) = (x,\cos{x})$. Am I right?

2. $A=\mathbb{R}$ and $B= \left\{ \langle x,y \rangle \in \mathbb{R}^2 | xy=4 \right\}$

Here I have some problems with $0$

3. $A=[0,+\infty)$ and $B$ is one of sides ($=a$) of regular pentagon

4. $A$ is Annulus $\mathbb{R} \left(r_1,r_2 \right)$ where $0<r_1<r_2$ and $B$ is rectangle $P_{a\times2a}$ for any $a\in\mathbb{R}$

5. $A$ is set of points of any circle and $B$ is set of solutions of $log_2(x-1)>0$ in $\mathbb{R}$

I think here I should put function like this one: $f(x)=(sin{x},cos{x})$ but I am not sure.

6. $A=\left\{ \langle x,y \rangle \in \mathbb{Q}^2 | y=x^2+3x+3 \right\}$ and $B=\left\{ x \in \mathbb{R} | \tan{x}=\sqrt{3} \right\}$

Answer 1

HINT: Your answer to the first question is the simplest of many correct answers.

From your comment about having a problem with $0$ it appears that you realize that you could answer the second question if you could find a bijection between $(0,1)$ and $[0,1)$; this answer suggests one way to do so.

The next three questions can all be answered using variations of that same basic idea. For instance, in $(4)$ you can slice the annulus along a radial line and open it up into a rectangle of the form $[a,b)\times[c,d]$. (Here I’m assuming that $A$ is the closed annulus. $B$ has the form $[a,b]\times[c,d]$. The idea from the previous question can be used to get a bijection between $(c,b)$ and $(c,b]$ for some $c\in(a,b)$, and this can then be extended to a bijection between $[a,b)$ and $[a,b]$. Finally, this can easily be extended to a bijection between $[a,b)\times[c,d]$ and $[a,b]\times[c,d]$. In $(5)$ you can break open the circle $A$ to get a bijection between it and $[0,1)$, and it’s not too hard to see find a bijection between $B=\{x\in\Bbb R:x>2\}$ and $(0,1)$.

The last question is a bit different. First find a bijection between $A$ and $\Bbb Q$; this is very straightforward. It’s also not hard to find a bijection between $B$ and $\Bbb Z$. To finish the job, you just need a bijection between $\Bbb Q$ and $\Bbb Z$, something whose existence you’ve probably already discussed.