In Terence Tao's book "Analysis I" 3rd edtion page 329 he wrote:
"• (Substitution axiom). Given any two objects x and y of the same type, if $x = y$, then $f(x)=f(y) $for all functions or operations $f$."
Here's my question: let $f(x)=\sqrt{x}$, and given we know $n^2=4$, then we yield $n=2$ from the Substitution axiom. However, we know that the solution for $n^2=4$ should be $n=2$ or $-2$ ! So where did I go wrong with substitution axiom, or is there any restriction with this axiom?
You don't quite yield $n = 2$; you yield $\sqrt{n^2} = 2$. That is true for both $n = 2$ and $n = -2$, just like the original equation.
The problem is that the square root function is not quite an inverse to the squaring function: that works only on nonnegative numbers. That is, the rule $$ \sqrt{a^2} = a $$ only holds for $a \geq 0$; if $a < 0$ we instead have the rule $$ \sqrt{a^2} = -a. $$ You can also summarize this as $\sqrt{a^2} = |a|$, in which case the substitution rule yields $|n| = 2$, again another way of saying that $n = 2$ or $n = -2$.