How to prove that $x=y=1$ or $x=y=−1$ if $xy=1$? I'm struggling with my current home assignment. If $x,y\in \mathbb{Z}$ and $xy=1$ how to prove that $x=y=1$ or $x=y=-1$? To achieve this, only the following properties can be used.
$\forall x,y,z\in \mathbb{Z}:$


*

*$(x+y)+z=x+(y+z)$;

*$x+y=y+x$;

*$x+0=x$;

*$x+(-x)=0$;

*$(xy)z=x(yz)$;

*$xy=yx$;

*$x\cdot 1=x$;

*$x(y+z)=xy+xz$;

*$xy=0 \implies x=0 \lor y=0$;

*$x<y \veebar x=y \veebar x>y$;

*$x<y \land y<z \implies x<z$;

*$x<y \implies  x+z<y+z$;

*$x<y \land z>0 \implies xz<yz$;

*$0 \neq 1$;

*$x+z=y+z \implies x=y$;

*$-(-x)=x$;

*$-(x+y)=(-x)+(-y)$;

*$x\cdot 0=0$;

*$z\neq 0 \land xz=yz \implies x=y$;

*$(-x)y=-xy=x(-y)$.
A hint would be greatly appreciated. 
As far as I understand, this is an exercise from this book: The Real Numbers and Real Analysis, Bloch, Ethan D.
 A: If $xy = 1$ with elements in a commutative ring, then $x,y$ are invertible in $R$.
In the ring of integers, the invertible elements are $\pm 1$.
A: First note that $x, y$ must be non-zero, and have the same sign.
Then note that if $x, y$ is a solution, so is $-x, -y$, as $x y = (-x) \cdot (-y)$. Thus we may assume $x, y > 0$.
Now note that if $x > 1$, then $1 = x y > y > 0$, but there are no integers between $0$ and $1$. 
A: You might try this: Let's prove it first, for $x>0,y>0$, later we'll prove the general case. Fix $x>1,x\in\mathbb{N}$ then we'll prove by induction $\nexists y\in\mathbb{N}, xy=1$, that is consider $A=\{y\in\mathbb{N}:xy>1 \}$
1)$1\in A$ Clearly, $x1=x>1$
2) Suppose $n\in A$, then $(n+1)x=nx+1>1+1>1$
Therefore, by the principle of induction $A=\mathbb{N}$
You have to prove now that if $x=1$ then the only solution is $y=1$, I believe the reasoning is very similar as above.
Now, clearly $x$ and $y$ have the same sign, otherwise if $x>0,y<0$ by the rules you said we can use $xy=yx<0<1$. So if $x<0\implies y<0$, choose $z=-x,z'=-y$ and you are in the above proved case.
Hopefully I made myself clear, otherwise feel free to ask and edit whatever you want. Have a nice day! :)
