Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
The notation means "The union over all $S$ in $C$ of $S$". – andybenji Dec 15 '12 at 11:12
up vote 5 down vote accepted

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 $A$.

share|cite|improve this answer

The definition of $\cup_{S \in C} S$ is: $\{x|\exists S\in C[x\in S]\}$

share|cite|improve this answer
Just to confuse the OP: This is often simply written as $\bigcup C$. – Hagen von Eitzen Dec 15 '12 at 11:19
yes. I saw this notation in my formal logic course. – Amr Dec 15 '12 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 $\bigcup_{S \in C} S = \emptyset \cup \{ \emptyset \} = \{ \emptyset \}$.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.