On axiom of choice page I have encountered this formula:
What does U-like symbol there means? It resembles union symbol, but union symbol should be used with two sets on the left and on the right. Then what does this symbol mean?
Union-like symbol meaning
elementary-set-theorynotation
Related Solutions
It depends on context, but I would say that you are unlikely to cause confusion if you write $A \sqcup B$ for the union of $A$ and $B$, where $A$ and $B$ are disjoint subsets of a larger space. For example, it is a fun fact about the $p$-adic numbers (denoted by $\mathbb{Q}_p$, where $p$ is some fixed prime number) that the unit ball in $\mathbb{Q}_p$ is composed of precisely $p$ congruent balls of radius $\frac{1}{p}$. Indeed, we could reasonably write $$ B(0,1) = \bigsqcup_{j=0}^{p-1} B(j, \tfrac{1}{p}).$$
Formally, as is noted in the question, the set $A\sqcup B$ is not the same thing as the set $A \cup B$. On the one hand, the union of two sets is defined to be $$A \cup B := \{ x : x\in A \lor x\in B \},$$ where $A$ and $B$ are subsets of some (thus far unspecified) universal set. On the other hand, $$A \sqcup B := (A\times\{0\}) \cup (B\times\{1\}), $$ where $X\times Y$ denotes the Cartesian product and $\cup$ is the union in the first sense. Note that the choice of $\{0\}$ and $\{1\}$ is arbitrary. Really, we just need two distinct elements: one to pair with elements of $A$, and another to pair with elements of $B$.
However, if $A, B \subseteq \mathscr{U}$ and $A \cap B = \emptyset$ (that is, if $A$ and $B$ are disjoint), then there is a correspondence between the union ($\cup$) and the disjoint union ($\sqcup$): define the function $f : A\sqcup B \to A\cup B$ by $$ f(x,k) := x $$ It is not too hard to see that this function is surjective: if $y \in A \cup B$, then either $y \in A$ and so $(y,0) \in A\sqcup B$ is mapped to $y$ by $f$, or $y \in B$ and so $(y,1) \in A\sqcup B$ is mapped to $y$ by $f$. Injectivity is similarly simple to show.
From a purely set-theoretic point of view, the two sets are isomorphic, i.e. $$ A \cup B \equiv A \sqcup B. $$ With a bit more work, we can show that this same map preserves other structures (under the right hypotheses). For example, if $A$ and $B$ are subsets of a topological space, then there is a natural topology on the set $A \sqcup B$ such that the disjoint union is homeomorphic to the union.
I checked the file with my pdf-viewer and got a message that this font cannot be displayed and is replaced. Then $10\%$ looked like $10\bullet\bullet$. It might be that your viewer displays it as $10 H$.
Best Answer
It is a union symbol. It is the union of all sets $S_i$, as $i$ ranges over all elements of the index set $I$.
$$\bigcup_{i\in I} S_i =\Bigl\{x\,\Bigm|\, \text{there exists }i\in I\text{ such that }x\in S_i\Bigr\}.$$
The symbol you describe is a special case, $$A_1\cup A_2 = \bigcup_{i\in\{1,2\}} A_i.$$