For any $X\subset \mathbb{R}$, is $|X|<|\mathbb{R}|, |X|\leq|\mathbb{R}|,$ or $|X|=|\mathbb{R}|$? So take for example $(20,35) \subset \mathbb{R}$
$(20,35)$ has an infinite number of elements and is uncountable
$\mathbb{R}$ has an infinite number of elements and is uncountable
Can we find a function $f:(20,35)\rightarrow \mathbb{R}$ that maps every element of the domain to the codomain?
Or is the infinity posed by $|\mathbb{R}|$ bigger than the infinity posed by $|(20,35)|$?
Like, for example, we know that $|\mathbb{N}|&lt;|\mathbb{R}|$ since for any $n \in \mathbb{N}$, we can find an $x \in \mathbb{R}:n&lt;x&lt;n+1$, thus $|\mathbb{N}|&lt;|\mathbb{R}|$
I would like to think that since $(20,35)\subset\mathbb{R}$, that by design, $|(20,35)|&lt; |\mathbb{R}|$ but the uncountability of the reals makes this dangerous to assume.
&quot;Can we find a function f:(20,35)→R that maps every element of the domain to the codomain?&quot; Yes.
If $g(x) = \frac {x-20}{15}$ then $g:(20,35)\to (0,1)$ is a bijjection.
And if $h(x)=\pi x - \frac \pi 2$ then $h:(0,1)\to (-\frac \pi 2, \frac \pi 2)$ is bijection.
And $\tan:(-\frac \pi 2,\frac \pi 2) \to (-\infty, \infty) = \mathbb R$ is a bijection.
So if $f= tan(\frac{x-20}{15}\cdot \pi - \frac \pi 2)=\tan(h(g(x))) $ the $f:(20,35)\to \mathbb R$ is a bijection.
SO $|(20,35)| = |\mathbb R|$.
By definition, a set is infinite if and only if it contains a proper subset with the same cardinality as itself. Put another way:
A set $K$ is infinite $\iff \exists L \subset M, L \neq M : |L|=|M|$
We can see examples of this with $2\mathbb{Z} \subset \mathbb{Z}$
We can find a bijective function $f: 2\mathbb{Z} \rightarrow \mathbb{Z}$ so that every element of $2\mathbb{Z}$ gets mapped to an element in $\mathbb{Z}$
In the same manner, with the uncountable reals,
For any nonempty proper subset $X \subset \mathbb{R},$ where $X$ is not countable,
Then $|X|=|\mathbb{R}|$
Here, also we can find a bijective function $f: X \rightarrow \mathbb{R}$ so that every element of $X$ gets mapped to an element in $\mathbb{R}$