5:26 PM
Can we prove that $\mathbb{R}\times\mathbb{R}$ is uncountable by a proof of contradiction and using an explicit function?
For example, suppose $\mathbb{R}\times\mathbb{R}$ is countable, that means there exists a function $f:\mathbb{R}\times\mathbb{R}\to\mathbb{N}$ which is injective.
Let $f((m,n))=2^m3^n\in\mathbb{N}$ where $m,n\in\mathbb{R}$.
since $2^m3^n$ is a natural number (as a product of two numbers) then each component is also a natural number, so $2^m,3^n\in\mathbb{N}$.
But then this implies that there exists an injective function $g:\mathbb{R}\to\mathbb{N}$ given by $g(m)=2^m$. Clearly this is a contradiction since $\mathbb{R}$ is uncountable.