Reflection of a set In analysis, we define a reflection of a set, say $E$ such that $E \subseteq \mathbb{R}$, as follows:
$-E := \{x : x = -a  \ \text{for some} \ a \in E\}$
So for example, $-(1, 2] = [-2, -1)$. 
My question is why does the definition say "for some"? To me, "for all" makes much more sense. 
 A: Note that $x\in -E$ if only if there is $a\in E$ such that $x=-a$. Now if there is $b\in E$ such that $x=-b$ then it implies that $a=b$. Therefore if in the definition say "for all", then the set $E$ has only one or zero element. Moreover "for some" means that "there is".  
A: An individual value $x$ will be equal to a particular $-a$: hence the use of the word some.  The word any might also work.
You want the set made up of all such $x$  where $a \in E$, but (assuming $E$ has more than one element) no individual $x$ will be equal to $-a$ for all the $a \in E$.  
A: For $E \subset \mathbb{R}$ let
$E' = \{x \in \mathbb{R} : (\exists a \in E)(x=-a) \}$
and
$E'' = \{x \in \mathbb{R} : (\forall a \in E)(x=-a) \}$
For example, $\{1,2\}' = \{-1,-2\}$ because, for the number $-1 \in \mathbb{R}$ there exists the element $1 \in \{1\}$ such that $-(-1)=1 \in \{1\}$ and for the number $-2 \in \mathbb{R}$ there exists the element $2 \in \{1,2\}$ such that $-(-2)=2 \in \{1,2\}$. But $\{1,2\}''= \varnothing$, because if there where some number $x \in \{1,2\}''$, the definition would require that for all element $a \in \{1,2\}$ we should have $x=-a$. So this implies that $x=-1$ and $x=-2$. Thus $-1=-2$, an absurd.
