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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?

Sunny88
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3 Answers3

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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.

Git Gud
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  • It is interesting that your answer is the opposite to what the people who commented on my question said. Do you think that they are mistaken? – Sunny88 Aug 18 '13 at 17:36
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    @Sunny88 Yes, they are mistaken. The only comment prevailing thus far implies that $x=\pm 1$ means that $x=1$ and $x=-1$, but that's not true. I've never seen it used like that. – Git Gud Aug 18 '13 at 17:38
  • Isn't it used like that when we say that $x=\pm 1$ is the solution to $x^2=1$ ? – Sunny88 Aug 18 '13 at 17:40
  • No, in that situation you conclude that $x=1$ or $x=-1$. Or instead of and being they key here. – Git Gud Aug 18 '13 at 17:40
  • The solutions to $x^2=1$ are $x=-1$ and $x=+1$. If $x^2=1$ then $x=-1$ or $x=+1$. Both are fine. – Fly by Night Aug 18 '13 at 17:43
  • " In mathematics, $\pm$ may indicate two possible values: one positive, and one negative." taken from here: http://en.wikipedia.org/wiki/Plus-minus_sign – Sunny88 Aug 18 '13 at 17:44
  • @Sunny88 Read what's below Written out in full, this states that there are two solutions to the equation: in the shorthand section. It's exactly what I said in my answer. – Git Gud Aug 18 '13 at 17:45
  • @FlybyNight You're mixing natural language with mathematics, by definition, $x=\pm 1$ is short for $x=1\lor x=-1$. – Git Gud Aug 18 '13 at 17:46
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    I think you are not reading the wikipedia article correctly. Nowhere there is it stated that $x = \pm 1$ stands for exactly $x=1∨x=-1$. The article is very careful to always say that this notation implies that both cases are the solutions. If $x = \pm 1$ meant exactly $x=1∨x=-1$, it wouldn't imply that both cases are solutions. – Sunny88 Aug 18 '13 at 17:56
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    @Sunn88 You write, I didn't read it correctly. Let me further explain then. Saying $-1$ and $1$ are solutions to the equation $x^2=1$, is the same as saying that the set of solution of $x^2=1$ is ${-1, 1}$, i.e., ${x\in \Bbb R\colon x^2=1}={-1, 1}$. Now take an arbitrary solution of the equation. Call it $x$,then $x=1\lor x=-1$ and we choose to abreviate the last formula by $x=\pm 1$. And of course $x=\pm 1$ is the same as saying that $1$ and $-1$ are solutions to the equation. – Git Gud Aug 18 '13 at 18:07
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    @Sunny88 But saying that $x$ is a solution to the equationis not the same as saying that $x=-1$ and $x=1$, rather it means that $x=-1$ or $x=1.$ – Git Gud Aug 18 '13 at 18:09
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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.

qaphla
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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?

Fly by Night
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