I have encounter this example in the notes, but not sure what did it mean.
$\cup_{S \in C} S = \emptyset \cup \{ \emptyset \}=\{\emptyset\}$ where $C= \{\emptyset,\{\emptyset\}\}$
Is this means union set "S" itself and "S" is in the "C" set?
I have encounter this example in the notes, but not sure what did it mean.
$\cup_{S \in C} S = \emptyset \cup \{ \emptyset \}=\{\emptyset\}$ where $C= \{\emptyset,\{\emptyset\}\}$
Is this means union set "S" itself and "S" is in the "C" set?
The notation $\bigcup_{S\in C}S$ means the union of all of the sets that are members of $C$. In this problem $C=\big\{\varnothing,\{\varnothing\}\big\}$, so as $S$ runs over the elements of $C$ it assumes just two values, $\varnothing$ and $\{\varnothing$. Thus,
$\bigcup_{S\in C}S=\underbrace{\varnothing}_{\text{when }S=\varnothing}\cup\underbrace{\{\varnothing\}}_{\text{when }S=\{\varnothing\}}=\{\varnothing\}\;,$
where the last step is because $\varnothing\cup A=A$ for any set $A$.
The definition of $\cup_{S \in C} S$ is: $\{x|\exists S\in C[x\in S]\}$
In general $\bigcup_{S \in C} S$ denotes the union of all sets belonging to the collection $C$, i.e., the collection of all objects that belong to at least one set in $C$.
If $C = \{ \emptyset , \{ \emptyset \} \}$, then $\bigcup_{S \in C} S = \emptyset \cup \{ \emptyset \} = \{ \emptyset \}$.