What is the difference between plus-minus and minus-plus? [duplicate]

Possible Duplicate:
What is the purpose of the $\mp$ symbol in mathematical usage?

Just as the title explains. I've seen my professor actually differentiating between those two. Do they not mean the same?

Solution 1:

If you write $$ \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b, $$ then + on the left side corresponds to minus on the right side, and - on the left side corresponds to + on the right side.

Standing alone, they mean the same.

Solution 2:

If it stands alone, say $a \pm b$, then it means the same as $a \mp b$. However, if they both occur in the same statement, such as $a\pm b \mp c$, then you may pick the "top" row or the "bottom" row of operators. In this case $a+b-c$ and $a-b+c$ would what is intended. But $a+b+c$ and $a-b-c$ would not be allowed.

Solution 3:

In general, we use $\pm$, but when we want to correlate a change of sign we also use $\mp$. For example: $2 (x \pm y) = 2x \pm 2y$, meaning that $2(x+y) = 2x + 2y$, and that $2(x-y) = 2x-2y$. Now, if we wanted the second sign to be the opposite of the first, we use $\mp$. For example: $-2(x \pm y) = 2x \mp 2y$ would mean that $-2(x+y) = -2x - 2y$ and $-2(x-y) = -2x + 2y$.

That is, whenever we have an expression involving $\pm$ or $\mp$, it's actually an abbreviation for two expressions: one in which we read all the top symbols ($+$ in $\pm$ and $-$ in $\mp$), and another one in which we read all the bottom symbols. Common examples:

$\sin (x\pm y) = \sin x \cos y \pm \cos x \sin y$ means $\begin{cases} \sin (x+ y) = \sin x \cos y +\cos x \sin y \\ \sin (x- y) = \sin x \cos y - \cos x \sin y \end{cases}$ $\cos (x \pm y) = \cos x \cos y \mp \sin x \sin y$ means $\begin{cases} \cos (x + y) = \cos x \cos y - \sin x \sin y\\ \cos (x - y) = \cos x \cos y + \sin x \sin y\end{cases}$

Now, when we don't have any changes of sign, like in $\sin (x\pm y) = \sin x \cos y \pm \cos x \sin y$ (all the top symbols are $+$), we could also write $\sin (x\mp y) = \sin x \cos y \mp \cos x \sin y$, and it would be the same, but this isn't common usage. The symbol $\mp$ only appears when there's already a $\pm$, but we want to establish a correspondence between opposite signs in an equation.

Note that it's only a matter of style; we could dispose completely of $\mp$ and used $\pm -$ instead, e.g., $\cos (x \pm y) = \cos x \cos y \pm (- \sin x \sin y)$.