I am trying to determine the truth value of the proposition "If $a^2 = b$ and $b > 0$, then $a=\sqrt{b}$.".
Based on the answer of my teacher, the truth value statement is false.
The counterexample is when $a=-\sqrt{b}$.
My answer is that the truth value of the statement is true.
We know that if $a^2 = b$ and $b > 0$, then $a=\pm\sqrt{b}$. Meaning $a$ can be positive OR negative square root of $b$.
If I will choose only one among the two possible conclusions (positive square root of $b$ OR negative square root of $b$), the statement will still be true.
Please let me know if my understanding is correct.
In mathematics, the meaning of a statement of the form
$$\text{If }A \text{ then }B$$
is
$$\text{If }A \text{ is true, then }B \text{ is } necessarily \text{ true}$$
You have interpreted it as
$$\text{If }A \text{ is true, then }B \text{ }might\text{ be true}$$
which is never a very useful thing to say.