Can we have a one-one function from [0,1] to the set of irrational numbers? Since both of them are uncountable sets, we should be able to construct such a map. Am I correct?
If so, then what is the map?

Define $f : [0,1] \to \mathbb{R}$ as follow : if $x \in \mathbb{Q}$, then $f(x) = \pi + x$. If $x$ is irrational, then $f(x) = x$. This function is injective. It is not a bijection.
