Proof that two sets have the same cardinality

Question

Look in the complex plane at the set $\mathbb{R}\cdot1\cup\mathbb{Z}\cdot i\subset\mathbb{C}$, the union of the real line and all whole multiples of $i$. Prove that this set has the same cardinality as $\mathbb{R}$. I'm having trouble finding a bijection. I would appreciate any help!

Answer 1

Consider $f :\left(\mathbb{R} \times \lbrace 0 \rbrace\right) \cup \left(\mathbb{Z} . i \right) \rightarrow \mathbb{R}$ defined for all $x \in \mathbb{R}$, $n \in \mathbb{Z}$ by $$ f(x,0) = \left\{ \begin{array}{ll} \frac{1}{2k-1} & \mbox{if } x=\frac{1}{k} \text{ for an integer }k\in \mathbb{Z}_{>0}\\ \frac{-1}{2k-1} & \mbox{if } x=\frac{1}{k} \text{ for an integer }k\in \mathbb{Z}_{<0}\\ x & \mbox{otherwise.} \end{array} \right. $$ $$ f(n.i) = \left\{ \begin{array}{ll} 0 & \mbox{if } n=0\\ \frac{1}{2n} & \mbox{otherwise.} \end{array} \right. $$

Show that $f$ is bijective.