1. ## Equinumerous Sets

1. Show that the following sets are denumerable (countably infinite)

(a) Q x Q (Q is the set of rational numbers) (This one is ok, but I am stuck at (b)

(b) Q( $\sqrt(2)$) = {a+b $\sqrt(2)$ | a $\in$ Q and b $\in$ Q}

2. Show that (0,1) ~ R (R is the set of reals)

3. Show that (0,1] ~ (0,1)

For 2 I simply could not think of the function from (0,1) to R that is 1-1 onto. For 3, the existence of {1} is a problem. I could not find a 1-1 function that can deal with {1}.

Thanks!

2. For 1b use what you know from 1a.
$\varphi :\mathbb{Q} \times \mathbb{Q} \mapsto \mathbb{Q}\left( {\sqrt 2 } \right)\;,\;\varphi (a,b) = a + b\sqrt 2$.
Show that $\varphi$ is a bijection.

For #2 consider the function $\tan \left( {\pi x - \frac{\pi }{2}} \right)$ on $(0,1)$.

For #3 define $F = \left\{ {\frac{1}{n}:n \in \mathbb{Z}^ + } \right\}$ for ease of notation.
Define a function $\Phi 0,1] \mapsto (0,1)\;,\;\Phi (x) = \left\{ {\begin{array}{rl}
{\frac{1}
{{n + 1}},} & {x = \frac{1}
{n} \in F} \\
x, & {\text{else}} \\

\end{array} } \right." alt="\Phi 0,1] \mapsto (0,1)\;,\;\Phi (x) = \left\{ {\begin{array}{rl}
{\frac{1}
{{n + 1}},} & {x = \frac{1}
{n} \in F} \\
x, & {\text{else}} \\

\end{array} } \right." />

3. ## Thanks

Thank you very much.

For question 1, I proceeded that way too,but I am stucked at showing f(a, b) is a 1-1 function. To show it, we have to set f(a1, b1) = f(a2,b2), but then we have only one equation, which I have no idea to show a1 = a2, and b1 = b2.