True or false: for all $x$ there is a $y$ such that if $x$ is non negative then $y^2 = x$ 
For all $x$ there is a $y$ such that if $x$ is non negative then $y^2 = x$

Is my logic correct in proving that statement is true ? Can provide an explanation of how to test this proof ?
$x=2$ then $y^2 = 2$
$x=2$ then $y = \sqrt 2$  
Truth table for this scenario
$t , t$
This proves there is $y$ such that $y^2 = x$


You cannot prove the statement by using only one value $x = 2$ to show that for that particular value $x$, there exists a $y$ such that $y^2 = x$. 
Why not?
You can't stop at showing it's true for one $x$, or two $x$, or even a million values of $x$, because the statement is a claim about all $x\geq 0$. Proving a "for all" statement requires a different strategy than proving an "existence" statement. It is true that for each $x$, there exists a $y$, and you can show this existence of $y$ by providing one value (only one is required for existence), but you need to prove that for every $x\geq 0$, some $y$ exists so that $y^2 = x$.
We usually do this by picking an arbitrary $x \geq 0$. By not assigning it any particular value, we keep it arbitrary, so that what we then demonstrate about $x$ holds for every $x\geq 0$.
So we take $x \geq 0$. Then $(\sqrt x)^2 = x$, so we can put $y = \sqrt x$,  and we're done, since no matter what the $x\geq 0$, $y = \sqrt x$ is defined, and hence exists.
