Translation of Union of Subsets of Vector Space

Theorem

Let $K$ be a field.

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

Let $\family {E_\alpha}_{\alpha \mathop \in I}$ be an indexed family of subsets of $X$.

Let $x \in X$.

Then:

$\ds \paren {\bigcup_{\alpha \mathop \in I} E_\alpha} + x = \bigcup_{\alpha \mathop \in I} \paren {E_\alpha + x}$

where $E_\alpha + x$ denotes the translation of $E_\alpha$ by $x$.

Proof

We have:

$\ds v \in \paren {\bigcup_{\alpha \mathop \in I} E_\alpha} + x$

if and only if:

$v = u + x$ for some $\ds u \in \bigcup_{\alpha \mathop \in I} E_\alpha$.

This is equivalent to:

there exists $\alpha \in I$ and $u \in E_\alpha$ such that $v = u + x$.

That is:

there exists $\alpha \in I$ such that $v \in E_\alpha + x$.

So by the definition of set equality, we have:

$\ds \paren {\bigcup_{\alpha \mathop \in I} E_\alpha} + x = \bigcup_{\alpha \mathop \in I} \paren {E_\alpha + x}$

$\blacksquare$