What is the Power Set of the Empty Set? | Set Theory

What is the power set of the empty set? We will answer this question in today’s math lesson! We will write the empty set like so: { }. Recall that the power set of a set A is the set containing all subsets of A. So, for example, P({ 1 }) = { { }, { 1 } }. Also, recall that if the cardinality of a set A is n, that is |A| = n, then the cardinality of the power set of A is 2^n, that is |P(A)| = 2^n.

So, since the empty set has 0 elements, we know |P({ })| = 2^0 = 1. There is one element in the power set of the empty set, and that one element is the empty set. So, P({ }) = { { } }. The power set of the empty set is the set containing the empty set because the empty set is the only subset of the empty set.

We know no other set can be a subset of the empty set because there are no elements in the empty set, so there is no set whose elements are entirely contained in the empty set except for the empty set itself. Again, the empty set is the only subset of the empty set, so the power set of the empty set is the set containing the empty set. That is, P({ }) = { { } }.

I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

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[patescalona]

This explanation was exactly what I was looking for! Nice video. Concise and just the right amount of background to help me tie some concepts together. Thanks!

[EsterPaulus-oz]

this is exactly what I have been failing to understand, thanks for the brief explanation

[nawarzarifeh]

loving this playlist! I needed a quick recap on some set theory definitions before diving into a new course. THANKS!

[nidhidodiya]

Thank You Very Much
It is really Very helpful for 11th..

[sneaksneak]

Given a set A = {} is A a subset of P(A)?
I think it is because the empty set is a subset of every set. Am I correct in thinking this ??
I keep finding conflicting answers online.
Thanks so much

[walala]

is there a difference between P({∅}) and P(∅) and what is it?

[vanshikasingh]

GREAT EXPLAINATION..WOULD U PLEASE HELP ME OUT FROM THIS QUESTION .... FIND THE NUMBER OF ELEMENTS IN ||||P(A)||||= ?, IF A=PHI

[samrahmariam]

So it means we can say that the power set of a empty set ia Singleton set right?

[biniamkifleyesus]

Excellent video!! Thanks But I still got that confusion about this empty set. Let's say we have a set X and we have the Power set of set X, P(X). Can we say {{}} is subset of P(X)?

[AustrianHungAryan]

From the textbook Mathematical Proofs: a Transition to Advanced Mathematics
There is a whole page of argumenting subsets and proper subsets.
But then you get this...
{Ø, {Ø}, {{Ø}}, {Ø, {Ø}}
Interestingly enough the comma happens 4 times for 4 set entries.
The Ø symbol occurs 5 times.
Buddha built his philosophy around his system of the 4 noble truths and the 5 aggregates. 4 and 5 give us 9. Buddha was alive circa the invention of 0 as a numeral. Voila Indian numerals.
0...
Ø
_

[Hilloftimber]

Great explanation man. Think i got it now :)

[thulasiammal]

Thank you for your clear explanation sir.

[kwongcc]

Clear Explanation! Thank you so much! Would like to confirm is { {} } = {} (Power set of Empty set equal to Empty set)?

[rashibhardwaj]

Subset of fi should be fi with curly but it is not so, why?

[Dailyfun-wn]

Sir, power set of set phi is also same as{phi} or {phi{phi}}

[TremorYt]

What is the power set of the null set ?
(either Null set or Singleton set or Both)
Pls....answer this ques

[ヽゝ-yj]

Hi if we had p(3) this is equal to {3} or equal to {∅, 3}?

[ahmedarman]

how many time have you said set, you are cool thanks

[amvrinn]

Is phi and {phi} the same thing? Please answer

[katherinevollrath]

I need to find the cartesian product of of the power set of an empty set and the power set of a nonempty set. Does this also result in an empty set like multiplying an empty set and a nonempty set?