About the meaning of the notation $\cup_{S \in C}$

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 means "The union over all $S$ in $C$ of $S$". Commented Dec 15, 2012 at 11:12

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: $\{x|\exists S\in C[x\in S]\}$

• Just to confuse the OP: This is often simply written as $\bigcup C$. Commented Dec 15, 2012 at 11:19
• yes. I saw this notation in my formal logic course.
– Amr
Commented Dec 15, 2012 at 11:21

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 $\bigcup_{S \in C} S = \emptyset \cup \{ \emptyset \} = \{ \emptyset \}$.