2

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.

  • 1
    Take your example, $E=(1,2]$. We agree that $-2\in-E$. Is it true that $-2=-a$ for some $a\in E$? Yes: $-2=-a$ if $a=2$ which is in $E$. Is it true that $-2=-a$ for all $a\in E$? Clearly not. For example $a=3/2$ is in $E$, but $-2\ne-3/2$. – David C. Ullrich Sep 12 '15 at 01:14

3 Answers3

0

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".

0

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 $a \in E$.

Henry
  • 157,058
0

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.

Gustavo
  • 2,016