What is the Power Set of the Empty Set? | Set Theory
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- čas přidán 4. 11. 2018
- 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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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!
Thanks for watching! I am glad it helped :) Let me know if you ever have any video requests!
loving this playlist! I needed a quick recap on some set theory definitions before diving into a new course. THANKS!
You're very welcome!
"set" counter = 92
Someone finally did it! I'll trust your count and thanks for watching. Wait until you see the next Power set lesson this week
Your channel is so helpful!
I'm glad you're finding the lessons helpful, thanks for watching!
Thank you for the clear explanation
You're very welcome, thanks for watching!
What if we do P(P(Φ)) then in theory I should get
Cardinal of Φ is 0
So no of subsets of Φ will be 2⁰ = 1
So the P(Φ)= 2¹= 2
So P(P(Φ))=2²=4
Thank you for your clear explanation sir.
My pleasure, thanks for watching!
Great explanation man. Think i got it now :)
Glad to hear it, thanks for watching!
GREAT EXPLAINATION..WOULD U PLEASE HELP ME OUT FROM THIS QUESTION .... FIND THE NUMBER OF ELEMENTS IN ||||P(A)||||= ? , IF A=PHI
Nice bro ,cleared my concept.
Glad it helped, thanks for watching!
Can someone please help with this? if aøb= 2a^2, what is the value of a if aø3 = 75?
is there a difference between P({∅}) and P(∅) and what is it?
Yes. Mainly because {ø} has a cardinality of 1 while ø has a cardinality of 0 and therefore are not equal.
You are a good explainer.
Thank you, I do my best!
Thank You Very Much
It is really Very helpful for 11th..
Glad it helped! Thanks for watching!
So it means we can say that the power set of a empty set ia Singleton set right?
Thanks for watching, Samrah, and that is correct! The power set of the empty set has only a single element, so it is a singleton set.
Good explanations of videos👍👏
I'm glad you found it helpful! Thanks for watching!
@@WrathofMath what's the difference between { } and { { }}
@@janewamuciimuiruri4444 {{ }} It Mean empty set as element of another set
{ } it mean empty set only
Hope you will understand
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
Clear Explanation! Thank you so much! Would like to confirm is { {} } = {} (Power set of Empty set equal to Empty set)?
Glad it was clear, thanks for watching! As for your question, I invite you to think about it for a minute. Do those two sets contain the same things? Once you've settled on your final answer, this video should help confirm: czcams.com/video/WgE5kr0Tj7M/video.html
@@WrathofMath Thank you! It helps a lot!
taking A shot every time i hear set
I love you so much man
i love you too
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)?
Thanks for watching! And think about the question step by step, considering definitions. To say { {} } is a subset of P(X) would mean that ever element of { {} } is an element of P(X). So, what are the elements of { {} }? And are they in P(X)? We could phrase this differently by remembering that P(X) contains the subsets of X. So then we are asking - is every element of { {} } a subset of X? If so, then indeed { {} } is a subset of P(X). What do you think?
@@WrathofMath Wow I get it now. Thanks man.
I cannot really get my head around this one. {{}} = P(∅), i.e. the power set of the empty set. By definition the empty set {} - not {{}} - is a subset of any set X. But the power set of the empty set P(∅) = {{}} is not necessarily a subset of P(X).
Is phi and {phi} the same thing? Please answer
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?
Thanks for watching, and have you tried writing it out? With the empty set, we have no element with which to make ordered pairs, which is why it's cartesian product results in an empty set. But the power set of the empty set is not empty, so we'll not have that problem.
@@WrathofMath I think I figured it out, thanks!
how many time have you said set , you are cool thanks
What is the power set of the null set ?
(either Null set or Singleton set or Both)
Pls....answer this ques
I’m not sure what you mean, that question is precisely the topic of this video. Did you watch through the video? Or am I misunderstanding your question?
Sir,power set of set phi is also same as{phi} or {phi{phi}}
Subset of fi should be fi with curly but it is not so, why?
Hi if we had p(3) this is equal to {3} or equal to {∅,3}?
Thanks for watching and for the question! P(3) is not defined because 3 is not a set.
@@WrathofMath thank you very much !!
Thanks for the video! Let me ask, if we had the formula
P(∅)={∅} (i),
correct, then previously we should have:
P(∅)=1 (ii),
and,
1={∅} (iii),
to through transitivity get (i).
While I doubt about the (iii). What implies it?
Thanks for watching and I am not sure about your question. What is meant by P(∅)=1? Are you talking about the cardinality of the power set of the empty set? Generally speaking, P(∅) is not 1 because 1 is not a set and P(∅) is a set. The only exception would be in the context of constructing the natural numbers using set theory, in which case we may be rigorously defining numbers as sets.
what would be the power set of the power set of empty set A (P(P(A))) ?
Empty set and {empty set}
Thanks bro.......
Glad to help!
Where are you from
D= {xlx=Ø} is this right ?
Wow
Explain what is a power set
What Is the number of subsets of the powerset of empty set?
1
Wouldn't it be right to say the empty set belongs to the power set of the empty set?
Yes indeed that is correct! One thing to take away from this is that there is no empty power set. So if we have a power set, we can surely consider at least one element from it!
@@WrathofMath Well stated.. thanks a ton sir! :)
@@WrathofMath Sir, I had some more doubts from elementary mathematics. If you don't mind, is there any way I can contact you personally for the same?
Is that symbol for empty set is an element for that set itself....? Can we call power set of empty set is Singleton set.
Thanks for watching and for the question! I’m not sure exactly which set you’re talking about but the empty set can be an element of a set. That’s right, the power set of the empty set is the set containing the empty set, thus it contains just one element, and is thus a singleton set.
What is the powerset of an empty set?
@@WrathofMath what's the difference between {{ }} and { }
@@janewamuciimuiruri4444 null set i mean empty set
@@janewamuciimuiruri4444 {{ }} It Mean empty set as element of another set
{ } it mean empty set only
Hope you will understand
do i undersatnd correctly that this is Creatio ex nihilo?
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
_
Are the power set of the empty set and the power set of the set containing the empty set the same !???
Please answer me
Thanks for watching and no they are not, hopefully this video helps! czcams.com/video/WgE5kr0Tj7M/video.html
Does ø = {ø}
Thanks for watching and for your question! Ask yourself "What does it mean for two sets to be equal?" It means that they have the exact same elements. More formally, if A and B are two sets, then A = B if and only if A is a subset of B and B is a subset of A. Also remember that the empty set isn't nothing, it is a set containing nothing. So putting the empty set in another set is not the same as adding 0 to a number. With all of that in mind, what do you think the answer is and why?
And if you have any other set theory questions, I am debuting a live show tomorrow at 7 pm EST where anyone can come and ask their math questions, so feel free to come by and ask any questions! Also I always welcome more questions and lesson requests here in the comments.
@@WrathofMath What's the difference between {{ }} and { }
Are u free for giving this type of knowledge more
1. Let A={{}, {{}}}. Find |P(A)| and P(A).
???
Но
Let's go!
"set"
Probably the word I've said more than all others.
100th comment
💯 🏆
Power set of empty set has how many subset...
so is n({0})=n(Ø) true
I do understand how it works, but still feels like a bullshit. What a contradiction.
Thanks for watching! I'm not sure what you're saying. Is there something I can help clear up?