Hey all, I can't seem to figure out the following proof:
Let x be an element of the Integers. If x*x = x then x=0 or x=1.
Any help would be appreciated!
Well it follows from the axioms of the integers that
$\displaystyle \begin{aligned} x\cdot x = x &\implies x\cdot x - x = 0\quad\text{\phantom{x.}(additive inverse)}\\ &\implies x\cdot(x-1)=0\quad\text{(distributive law)}\end{aligned}$
Since the integers form an integral domain (not sure how much abstract algebra you know), there are no zero divisors. So either $\displaystyle x=0$ or $\displaystyle x-1=0$.
Does this make sense?