# Prove $A= \mathbb{R}$

It can be seen easy but i'm really stack.I don't know how to start.

Let $A \neq \emptyset$ is a subset of $\mathbb{R}$.If for every real $x$,$y$ sum $x+y$ belongs to $A$,then $xy$ also belongs to the set $A$.Then prove that $A= \mathbb{R}$

$\mathbb{R}$- here is the set of real numbers. Thanks for help in advance!

• hi,i'm new here,i don't know how to use here,i will be thankful to you if you help me Sep 5, 2015 at 18:31
• Where you have considered $x,y$ in $A$ or in $\mathbb{R}$? If in A this is not true, if in $\mathbb{R}$, these assumptions are not enough to say that $A=\mathbb{R}$ Sep 5, 2015 at 18:56
• As I said for every real($R$) x,y Sep 5, 2015 at 18:59
• Your treatment of quantifiers is kind of sloppy. As it stands, $A$ can be any subset of reals (because either $A={\bf R}$ or there is some $x+y\notin A$. Sep 5, 2015 at 19:42

First, since $A\ne \emptyset$, there exists a $y=0+y\in A$, so $0\times y=0\in A$.
Then, for all $y\in\mathbb R$, $0=y+(-y)\in A$, so $y\times(-y)=-y^2\in A$, i.e. all negative real numbers are in $A$.
Finally, for all $y\in\mathbb R_{<0}$, $y=(y/2)+(y/2)\in A$, so $(y/2)(y/2)=y^2/4\in A$, i.e. all positive real numbers are in $A$.
As a result, $A=\mathbb R$.
Let $a\in A$ be an element. Then to show that $x\in\mathbb{R}$ is in $A$ (for $x$ arbitrary) amounts to solving the system $$\cases{y+z = a, &\\yz = x.}$$ Can you take it from here?