Dimension of vector space and cardinal number of sets have very similar properties :
- Let $U$ be a vector space and $V, W \subseteq U$ two vector subspace, so $$ \dim(V + W) = \dim(V) + \dim(W) - \dim(V \cap W), $$ if $V \cap W = \{0\} \; \Leftrightarrow \; \dim(V \cap W) = 0$, thus $$ \dim(V + W) = \dim(V) + \dim(W), $$ and we write $V \oplus W$, the direct sum of $V$ and $W$.
- Let $A$ be an arbitrary set and $B, C \subseteq A$ two subsets, so $$ \mathrm{card}(B \cup C) = \mathrm{card}(B) + \mathrm{card}(C) - \mathrm{card}(B \cap C), $$ if $B \cap C = \emptyset \; \Leftrightarrow \; \mathrm{card}(B \cap C) = 0$, thus $$ \mathrm{card}(B \cup C) = \mathrm{card}(B) + \mathrm{card}(C), $$ and… nothing !
So my question is : Is there an operation “direct union” which is to the union of sets what the direct sum is to the sum of vector space ?
I found an operation called disjoint union, but it is not (at all) what I am looking for (disjoint union is indexed union to keep information of the parent set).
Given that write $U = V \oplus W$ is much concise than $(U = V + W) \wedge (V \cap W = \{0\})$, it would be nice to have a notation for “direct union”, e.g. ${\,\bigcirc\mathrel{\mspace{-19.5mu}\cup}}$, such that $$ A = B {\,\bigcirc\mathrel{\mspace{-19.5mu}\cup}} C \quad\Leftrightarrow\quad (A = B \cup C) \wedge (B \cap C = \emptyset). $$
\coprod($\coprod$). – Bernard Jun 30 '20 at 08:45