- #1
Mr Davis 97
- 1,462
- 44
Homework Statement
##\mathbb{R} \setminus C \sim \mathbb{R} \sim \mathbb{R} \cup C##.
Homework Equations
The Attempt at a Solution
I have to show that all of these have the same cardinality. For ##\mathbb{R} \cup C \sim \mathbb{R}##, if ##C = \{c_1, c_2, ... c_n \}## is finite we can define ##
f(x) =
\begin{cases}
n &~ \text{if} ~x=c_n~ \text{where} ~n \le |C| \\
n+|C| &~ \text{if} ~x=n~ \text{where} ~n \in \mathbb{N} \\
x &~ \text{if} ~x \not\in \mathbb{N} \cup C
\end{cases}
##
And if ##C = \{c_1, c_2, c_3, ... \}## is infinite we can define
##
f(x) =
\begin{cases}
2n &~ \text{if} ~x=c_n~ \text{where} ~n \in \mathbb{N} \\
2n-1 &~ \text{if} ~x=n~ \text{where} ~n \in \mathbb{N} \\
x &~ \text{if} ~x \not\in \mathbb{N} \cup C
\end{cases}
##
I think that these are both bijections.
However, I am a little confused about showing that ##\mathbb{R} \sim \mathbb{R} \setminus C##. Does this imply that all the elements of ##C## are in ##\mathbb{R}##? If this is the case couldn't we just define ##f: \mathbb{R} \rightarrow \mathbb{R} \setminus C##, and have a very similar bijection as the first one defined above?