If, say, we know that $x=1$, is then the expression $x=\pm 1$ mathematically incorrect? I ask this because when we use $\pm$ sign in the discriminant formula we imply that any of the plus or minus cases is possible. But in this case only one case is possible. Is it wrong to use $\pm$ sign here? How can I convey the difference in meaning?
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$\begingroup$If $x=1$, then $x=\pm 1$ is not incorrect, that is, it is not false.
The notation $x=\pm a$ is not an equality, it isn't saying that $x$ and $\pm a$ are the same thing, in fact $\pm a$ has no meaning by itself, it doesn't represent anything. An expression such as $x=\pm a$ should be read as a whole, as a unique symbol and it is simply short hand notation for $x=a \lor x=-a$.
In this particular case, since $x=1$ is true, then so is $x=1 \lor x=-1$, which is to say $x=\pm 1$ is true.
$\endgroup$ 11 $\begingroup$If $x = 1$, then $x = \pm 1$ is correct, though you do lose information. $x = \pm y$ is syntactic sugar for $x \in \{ -y, y \}$, and so, since if $x = 1$, $x \in \{-1, 1 \}$, $x = \pm 1$ is valid.
$\endgroup$ $\begingroup$In my opinion, if you are asked to solve $x=1$ and you write $x=\pm 1$ then your answer is wrong. I can see that there are logical arguments to the contrary, but they seem to ignore the spirit of the notation.
It is a fact that $x^2=1 \iff x = \pm 1$. It is false to write $x=1 \iff x = \pm 1$.
When listing solutions to equations, the equivalence is a tacit assumption. In other words, people don't list redundant values; it might be logically correct to say $x^2=1 \implies x \in \mathbb{C}$, but in reality, what use is that?
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