Disjoint Union Preserves Domination

Theorem

Let $I$ be an indexing set.

For all $i \in I$, let $A_i$ and $B_i$ be sets such that $A_i \preccurlyeq B_i$.

Here, $\preccurlyeq$ denotes domination.

Then:

$\displaystyle \bigsqcup_{i \mathop \in I} A_i \preccurlyeq \bigsqcup_{i \mathop \in I} B_i$

where $\bigsqcup$ denotes disjoint union.

Proof

By definition of domination, for all $i \in I$, there exists an injection $\iota_i: A_i \to B_i$.

Thus the mapping $\displaystyle \iota : \bigsqcup_{i \mathop \in I} A_i \to \bigsqcup_{i \mathop \in I} B_i$ defined by:

$\iota \left({x, i}\right) = \left({\iota_i \left({x}\right), i}\right)$

is an injection.

$\blacksquare$