Sum of Union of Subsets of Vector Space and Subset

Theorem

Let $K$ be a field.

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

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

Let $C \subseteq X$.

Then:

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

Proof

We have:

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

if and only if there exists $c \in C$, $\alpha \in I$, $x \in E_\alpha$ such that:

$v = c + x$

That is, if and only if $v = c + x$ for $c \in C$ and:

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

This is equivalent to:

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

so, we get:

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

by the definition of set equality.

$\blacksquare$