Translation of Complement of Set in Vector Space

Theorem

Let $K$ be a field.

Let $X$ be a vector space over $K$.

Let $F \subseteq X$ be a non-empty set.

Let $x \in X$.

Then:

$X \setminus \paren {F + x} = \paren {X \setminus F} + x$

It is immediate that if $y \in F$ we have $y + x \in F + x$.

Conversely, if $y + x \in F + x$, then $y + x = u + x$ for some $u \in F$.

That is, $y = u$ and so $y \in F$.

Hence we have $y \in F$ if and only if $y + x \in F + x$.

Hence for $y \in X$ we have $y + x \not \in F + x$ if and only if $y \not \in F$.

That is, $y + x \in X \setminus \paren {F + x}$ if and only if $y \in X \setminus F$ if and only if $y + x \in \paren {X \setminus F} + x$.

So $y + x \in X \setminus \paren {F + x}$ if and only if $y + x \in \paren {X \setminus F} + x$.

We can therefore conclude:

$X \setminus \paren {F + x} = \paren {X \setminus F} + x$

$\blacksquare$