Let $X$ be the set of real transcendental numbers. Define the realtion $\sim$ on $X$ by $x\sim y$ iff $x-y \in \mathbb{Q}$ is an equivalence relation.
Let $Y$ denote the set of equivalence classes generated by $\sim$ defined above.
Define the function $P:X \rightarrow Y$ by $f(x) = [x]$ where $[x]$ denotes the equivalence class of $x$. Does the function P have:
- a left inverse?
- a right inverse?
My work so far:
Since $P$ is surjective by definition of equivalence classes, $P$ has a right inverse.
Now if $P$ is injective then it will have a left inverse. I think it's not injective as two real transcendental numbers can end up in the same equivalence class, so $P$ doesn't have a left inverse, but I am not sure how to show this rigorously.
Hint 1: prove that if $x\in X$, then also $x+1\in X$. Since $x+1\sim x$, the map $P$ is not injective
Hint 2: the sum of algebraic numbers is algebraic.