TITLE: If $x^2=x$, then $x=1$. Is this statement true or false?
QUESTION [12 upvotes]: I understand that this equation has two solutions, that is $x=1$ or $x=0$. But if you say this statement is false, you are like saying that $x=1$ is not the solution for the equation. If a basket contains $2$ apples and $1$ orange, and I say that the basket contains $2$ apples. Is there anything wrong? Anyone have a good argument one way or the other?

REPLY [2 votes]: A part from the logical problem of reversing implications:
$$x=1\Rightarrow x^2=x\qquad \text{is different from}\qquad x^2=x\Rightarrow x=1$$ 
If you don't specify the nature of $x$, than the claim is largely false in general, as it is not true that $x^2=x\Rightarrow x=0,1$. On the other hand, there are algebraic structures where the claim is true.
Example. 
Let $f:\mathbb R^3\to\mathbb R^2$ be the projection on the $XY$-plane along the $Z$-axis.
Then, for every $p\in\mathbb R^3$ you have $f(f(p))=f(p)$. Thus $f^2=f$. Nonetheless $f$ is different from both $1$ (the identity) and $0$ (the zero map.)
When the claim it is true:
If you are in a Group, where there is a unique identity and all elements are invertible, then by the cancellation law (that say $ax=bx\Rightarrow a=b$) you have $$x\cdot x=1\cdot x\Rightarrow x=1$$