Bijective map from a set to a subset of reals? There is a concept that I have been thinking about quite a lot lately as I am currently self-studying point-set topology:
Say we have a bijective map from one interval, $[a,b]$, to another interval, $[c,d]$, both of which are in $\mathbb{R}$. Also set $c$ and $d$ so that $[c,d] \subseteq [a,b]$.
How can it be that function maps to a subset which is a proper subset of the map's preimage bijectively? I.e. How can the map be both one-to-one and onto when the image should contain &quot;less&quot; elements than the domain?
One example would be $f(x) := \frac{x}{1+x}: [0,10] \to [0, \frac{10}{11}]$
I'm hoping someone can show me why this isn't such a strange concept? Is there a theorem or result that explains this or provides some intuition?
The same applies to any infinite set. In fact &quot;infinite set&quot; can can be defined as a set that contains a proper subset with the same cardinality. More precisely these are known as Dedekind-infinite sets.
For example take naturals $\mathbb{N}=\{0,1,2,3,\ldots\}$ and its proper subset $\mathbb{N}_+=\{1,2,3,\ldots\}$ and note that there's a simple bijection between them $x\mapsto x+1$. You can even remove infinitely many elements from $\mathbb{N}$ and still end up with the same cardinality, e.g. for $2\mathbb{N}=\{0,2,4,6,\ldots\}$ we have a bijection $x\mapsto 2x$ even though there are infinitely many elements in $\mathbb{N}\backslash 2\mathbb{N}$.
And so &quot;being a subset&quot; and &quot;being equinumerous&quot; are loosely related concepts. At most we know that $|A|\leq |B|$ when $A\subseteq B$. But $A\subsetneq B$ doesn't imply $|A|&lt;|B|$, unless $B$ is finite.

I'm hoping someone can show me why this isn't such a strange concept?

The idea may be strange to you. Infinities are weird. But most people simply accept that and move on. There's not really anything more to do about it. After some time you get used to it and it becomes a simple fact of mathematical reality.
Consider as an example there exists a bijection between $\mathbb{Z_+}$ and $\mathbb{Z}$.  While these are subsets of $\mathbb{R}$, the concept is the same.
Let $f: \mathbb{Z} \to \mathbb{Z_+}$ defined by
$f(i) = \{1$ if $i = 0$; $2i$ if $i$ is positive; $1-2i$ if $i$ is negative