$\forall a,b \in {\mathbb R^+}$: if $a=b$ then $\sqrt a =\sqrt b$ Is this statement:
$\forall a,b \in {\mathbb R^+}$: if $a=b$ then $\sqrt a =\sqrt b$
True or False? If False, then how?
I know we could take $a = b = 9$ and then continue with $ 9 = (-3)^2 = 3^2$ and try that. But my professor said in class that the negative sign arises 'externally' or when we use the convention $\pm4$ when square rooting a number, say, $16$ it comes from outside. This has got me pretty confused.

The square root is a function from $[0,+\infty[$ to $[0,+\infty[$. For any function $f$, if $a=b$, then $f(a)=f(b)$ (provided $f$ is defined at $a$).
Your professor is probably referring to the situation where you look for solutions of the equation $x^2=a$ with $a\ge0$, and then there are two solutions, $+\sqrt{a}$ and $-\sqrt{a}$. But the sign is not "part" of the function $x\to\sqrt x$.
