# Thread: Super set confusion

1. ## Super set confusion

let $\displaystyle P={\{\emptyset,1\}}$ be the given set.

What are elements present in the set $\displaystyle P \cup S$, where $\displaystyle s$ is super set of P

2. Originally Posted by kumaran5555
let $\displaystyle P={\{\emptyset,1\}}$ be the given set. What are elements present in the set $\displaystyle P \cup S$, where $\displaystyle s$ is super set of P

$\displaystyle S$ is superset of $\displaystyle P$ iff $\displaystyle S\supset{P}$ so, $\displaystyle P\cup S=S$

Fernando Revilla

3. Originally Posted by kumaran5555
let $\displaystyle P={\{\emptyset,1\}}$ be the given set. What are elements present in the set $\displaystyle P \cup S$, where $\displaystyle S$ is super set of P
I am guessing that there is a translation error here.
Can it be that super set should be power set, $\displaystyle \mathcal{P}(P)$.

If that is correct then $\displaystyle S=\mathcal{P}(P)= \left\{ {\emptyset ,\{ \emptyset \} ,\{ 1\} ,P} \right\}$.

So, I think the question asks for $\displaystyle P\cup \mathcal{P}(P)$ which is a standard question is a basic set theory class.

4. You are correct. It was my mistake typing it as super set.

what is difference between $\displaystyle {\emptyset}$and $\displaystyle {\{\emptyset\}}$

I wrote the answer with three elements by combining both of them.

Please explain me the difference.

5. Originally Posted by kumaran5555
$\displaystyle {\emptyset}$and $\displaystyle {\{\emptyset\}}$

Please explain me the difference.

$\displaystyle {\emptyset}$ is the emptyset itself while $\displaystyle {\{\emptyset\}}$ is the set containing the empty set ... I'm not sure why it lists both seperately however even my texts do the same ... I've learned to just remember it when it comes to the power set.

6. It is very important to understand that the above two are different. Think about how will you "define" $\displaystyle {\emptyset}$ ?
Your definition should be precise (basically you cannot say that it contanins nothing)

Once you do that you will understand the difference betweem the above.

7. As far as i know, it is defined as a set with zero elements.

8. Alright -
So now can you see the difference between

$\displaystyle {\{\emptyset\}}$

and

$\displaystyle {\emptyset$

How many elements does each have?

PS: Though I am no expert on this subject but I feel $\displaystyle {\emptyset$ has deep rooted significance - for e.g. you can question what do you mean by a set with 0 element? Maybe someone with more relevant knowledge can comment. But the two sets you mentioned are definately very different and it would be a blunder to consider they are same.

9. Okay.

$\displaystyle \emptyset$ has zero elements.

$\displaystyle {\{\emptyset\}}$ has one element which has zero elements :-)

10. So to state it better - $\displaystyle {\{\emptyset\}}$ has 1 element - And that element in turn is a set which has zero elements. sorry if i'm being stingy !!

11. A note on why this topic is included in basic set theory.

The set $\displaystyle P=\{\{a\},\{a,b\}\}$ contains two elements.
Both of the elements are sets. Neither a nor b is an element of $\displaystyle P$.
Because $\displaystyle a\ne \{a\}~\&~ b\ne \{a,b\}$
But we use the set $\displaystyle P$ to define the ordered pair $\displaystyle (a,b)$.

12. Originally Posted by Plato
The set $\displaystyle P=\{\{a\},\{a,b\}\}$ contains two elements.
Yes, if a does not equal b. Otherwise, it has one element, namely {a}.