What is Power set: Definition and 47 Discussions

In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set.
The powerset of S is variously denoted as P(S), 𝒫(S), P(S),


{\displaystyle \mathbb {P} }
(S), ℘(S) (using the "Weierstrass p"), or 2S. The notation 2S is used because given any set with exactly two elements, the powerset of S can be identified with the set of all functions from S into that set.Any subset of P(S) is called a family of sets over S.

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  1. pellman

    I Why is the Axiom of Power Set needed?

    In the Zermelo-Fraenkel axioms of axiomatic set theory we find: Axiom. Given any set x, there is a set such that, given any set z, this set z is a member of if and only if every element of z is also an element of x. Why is this needed as an axiom? why isn't it merely a definition? Under...
  2. T

    I Axiom of Choice: Disjoint Family ##\Rightarrow ## Power Set

    So apparently the proof involves a trick that converts the problem of a general power set ##\mathscr{P}(M)## of some set ##M## which has of course the property of not having pairwise disjoint set-elements to a problem that involves disjoint set-elements. I do not understand why this trick is...
  3. PengKuan

    Cardinality of the set of binary-expressed real numbers

    Cardinality of the set of binary-expressed real numbers This article gives the cardinal number of the set of all binary numbers by counting its elements, analyses the consequences of the found value and discusses Cantor's diagonal argument, power set and the continuum hypothesis. 1. Counting...
  4. Gh. Soleimani

    I Are These Rules new Conjectures in Set Theory?

    We can easily find out below rules in set theory: 1. Let consider set “A” as follows: A = {a1, a2, a3, a4… an} and also power set of A is set C: C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}} Rule 1: To find the number of subsets with precise members number, we can use Binomial...
  5. T

    Proof of |2^N x 2^N| = |2^N| with N the natural numbers

    Hello, At my exam I had to proof the title of this topic. I now know that it can easily be done by making a bijection between the two, but I still want to know why I didn't receive any points for my answer, or better stated, if there is still a way to proof the statement from my work. My work...
  6. S

    Is the Subtraction of Power Sets Possible?

    I have two quick questions: With P being the power set, P(~A) = P(U) - P(A) and P(A-B) = P(A) - P(B) I'm told if it's true to prove it, and if false to give a counterexample. To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get...
  7. H

    Understanding the Power Set of a Set X: Proving Its Existence | Homework Help

    Homework Statement Let X be a set. Then the set {Y:Y is a subset of X} prove this is a set.Where do i start? Really unsure, i know that i have to use the power set? I have written down; {0,1}^X
  8. C

    How do you calculate the power set of a set of sets?

    How are you supposed to go about putting together the power set of a set of sets such as X = {{1},{1,2}} What is the power set of X then? And what's the rule for calculating cardinality for the power set of a set that consists of elements which are sets such as the above? Because the set X...
  9. C

    Finding Cardinality of Power Set

    Homework Statement Let S be the set of functions from a set A to {0,1} Prove that |P(A)|= |S| Homework Equations P(A) is the power set of A The Attempt at a Solution I have no idea how to do this... If A is finite then A has n elements, and we can write out the elements from one to...
  10. S

    Same power set implies set equality

    Homework Statement Can you conclude that A = B if A and B are two sets with the same power set? Homework Equations The Attempt at a Solution I know intuitively that A and B have to be equal, because all the individual entities in the power set (you know what I mean) have to be in...
  11. T

    Is Writing ##2^{\mathcal{A}}## an Abuse of Notation for the Power Set?

    I am confused I understand that the power set has ##2^{|\mathcal{A}|}## members, but they write it as ##2^{\mathcal{A}}## I don't understand why they just don't write it as ##\mathcal{P}(\mathcal{A})## to refer to the power set which has ##2^{|\mathcal{A}|}## elements, isn't that an abuse of...
  12. D

    Help With Find The Cardinality of a Power Set of a Cartesian Product

    Homework Statement Suppose that A and B are finite sets. What is |P(AxB)|? Meaning what is the cardinality of the power set of a cartesian product of the sets A and B. Homework Equations |AxB|=|A| * |B| since A and B are finite sets Power set of a set is the set of all subsets of...
  13. caffeinemachine

    MHB An increasing function on the power set of a set

    Let $A$ be a finite set. Let $\mathcal{P}(A)$ denote the power set of $A$. Let $f:\mathcal{P}(A)\to \mathcal{P}(A)$ be a function such that $X\subseteq Y\Rightarrow f(X)\subseteq f(Y)$. Show that $\exists T\in \mathcal{P}(A)$ such that $f(T)=T$. P.S. The power set of $A$ is the set of all the...
  14. N

    Stuck on a Power Set Problem - What's the Lacking Set?

    While practicing power set problems I came across one that has me stumped. The problem asks: Is the following set is a power set of of a set? {∅, {b, ∅}, {a}, {a, b}, {b}} My answer: This set has 5 elements. Since 5 is not a power of 2, this cannot be the power set of any set. The...
  15. B

    What Are the Elements of P(P(∅))?

    Homework Statement P(P(∅)) Homework Equations The Attempt at a Solution I did the inner power set first: {{∅}, ∅} Then I took the power set of that: {{{∅}}, {∅}, {{∅}, ∅}, ∅} But according to the answer, there is only two elements.
  16. B

    Finding The Power Set Of A Given Set

    Homework Statement {∅,{∅}} Homework Equations The Attempt at a Solution My answer is {∅, {{∅}}, {∅, {∅}}} but the actual answer is: {∅,{∅},{{∅}},{∅,{∅}}} I don't understand how the second element, {∅}, appears...
  17. K

    Power Set Unions and inclusion

    I am studying Velleman's "How to prove it: a structured approach" and I have to say that it is one of the best decision I have taken. Right now I am working on the exercises that are on Velleman's page along with the java applet "Proof Designer" and I have the feeling I start to get a bit how...
  18. M

    Does ZFC Imply the Power Set of Naturals?

    Is it true that for every standard formulation T of ZFC, T ⊢ the power set of {naturals}? After all, the empty set axiom and the pairing axiom are in T, and so we get N. Then by the power set axiom we get P(N).
  19. J

    The power set of the power set of an infinite set

    Let X be a set which is countably infinite. Then is there any example, on earth, of the power set of the power set of X?
  20. M

    Power set P(S) with symmetry difference.

    Homework Statement Determine the orders of all the elements of the power set P(S) of a set S with symmetric difference Δ. Homework Equations The Attempt at a Solution If A,b are two elements of the power set the symmetric difference is AΔB = (A-B) U (B - A) How are we...
  21. C

    Exploring the Power Set: \aleph_0 and Beyond

    If I started with a one element set and took its power set. And then I just kept taking the power set forever, would I eventually end up with a set that had cardinality of \aleph_0 ?
  22. J

    What Is the Power Set of the Empty Set?

    Homework Statement Is this true or false Homework Equations {{ø}} is a subset of P({ø,{ø}}) The Attempt at a Solution I'm thinking that it is since to make the subset {ø} into a power set it becomes {{ø}}?
  23. J

    Is This the Correct Power Set for X={S,{S}}?

    Homework Statement Is this the right power set for x? Homework Equations X={S,{S}} The Attempt at a Solution P(x)={ø, {S}, {{S}}, {S,{S}}}
  24. G

    Cardinalities of power set vs B^A

    Hi everyone, If B^A is the set of functions mapping from A \rightarrow B = \{ 0, 1 \}, prove that |B^A| = |P(A)|, where P(A) is the power set of A. Is it as simple as letting the mapping from B to A be denoted by \phi and defining a_1, a_2 \in A, a_1 \ne a_2 such that \phi (a_1) = 0 and...
  25. I

    Proving the Equivalence of Power Sets: A Challenge in Bijection

    Hi Here is the problem I am doing. Prove that ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+})\sim \mathcal{P}(\mathbb{Z^+}) Where ^{\mathbb{Z^+}}\mathcal{P}(\mathbb{Z^+}) is the set of functions from \mathbb{Z^+} to \mathcal{P}(\mathbb{Z^+}). To prove this I will need to come up with...
  26. T

    Proving Bijective Power Sets of A & B | A, B, C

    Let A,B, and C be non-empty sets. A and B are bijective. Prove that the power set of A is bijective to the power set of B. I understand how to prove bijection but can't figure out how to apply this to power sets and can't find any info on this subject.
  27. S

    Bijection for a power set function

    Homework Statement Let N={1,2...n} .Define the Power set of N,P(N). a) show that the map f:P(N)->P(N) defined by taking A to belong to P(N) to N\A is a bijection. b)C(n,k)=C(n,n-k). The Attempt at a Solution Now the power set is defined by P(N)=2^n and a bijection is a one-to-one...
  28. S

    Proving Power Set Notation: Let B \subseteq U

    my actual problem is to let B be a subset of the set U and prove P(B^{C}_{U}) \neq (P(B))^{C}_{P(U)} but I am confused on the scripts and not quite sure what they are wanting me to do i have Let B \subseteq U where B = {b} and U = {B} I know P(B) = {empty set, {b}} and P(U) = {empty...
  29. M

    Why Does the Power Set of a Set Have 2^n Elements?

    Homework Statement Why is the size of the power set 2^n ? To elaborate, suppose we have a set B that has n elements. Let C be the set that contains all the possible subset of B. Why is it that |C|=2^n ? It boggles my mind why the base is 2 for all size of sets. Thank you, M...
  30. A

    Set Theory: Subsets & Power Set of A - Joe

    I am currently covering Set Theory from the book, A Transition to Abstract Mathematics (Douglas Smith) and have a question about subsets and an implication. The statement reads as follows: If B is a subset of A, then {B} is an element of the Power Set A. I choose this to be true. By...
  31. S

    How Does the Power Set Size Double with Each Additional Element?

    Homework Statement Let [n] denote the set consisting of the first n natural numbers 1, 2, 3, ..., n. Use induction to prove that the power set P([n]) has exactly 2^n elements. The Attempt at a Solution 1) Base case: P([1]) = {{1}, null}. 2^1 = 2 elements. 2) Inductive step: Assume P([n]) has...
  32. H

    Prove Power Set equation by Induction

    Homework Statement (a) Use induction to show that if n(A) = n then n(P(A)) = 2^n. n(A) is the cardinality of set A. P(A) is the power set of A.Homework Equations The answer is We apply induction to prove the claim. If n = 0 then A = null and in this case P(A) = {null}. Thus n(P(A)) = 1 =...
  33. T

    Power set (set theory) question

    I'm reading a book about set theory and it introduced the concept of power set. Ok, I understand what is a power set and the entire concept but I have a question about the number of elements of a power set. There's written in the book that the number of elements of a power set is 2n where n...
  34. T

    Why is P(A) called the power set of A?

    Why is P(A) called the power set of A? I don't know what to say about this... can you explain this to me?
  35. J

    Why does \mathcal{P}(X \cup Y) not always equal \mathcal{P}X \cup \mathcal{P}Y?

    It seems intuitive that the power set of a union of sets P(XunionY) is not a subset of the union of the two respective power sets P(X)unionP(Y). For finite sets the former will have more elements than the latter. However, I can't figure out what is wrong with the following line of reasoning...
  36. J

    Obtain the power set p(s) if S(a,b,c)

    What is power set?Obtain the power set p(s) if S(a,b,c)
  37. nicksauce

    Proving f is Not Surjective: A Finite Set Mapping to its Power Set

    Homework Statement Suppose that f is a mapping from a finite set X to P(X) (the power set of X). Prove that f is not surjective.Homework Equations The Attempt at a Solution My proof strategy is 1) Show that P(X) always has more elements than X 2) Show that a mapping from X->Y cannot be...
  38. E

    Onto Mappings from a Set to Its Power Set

    Homework Statement Prove that there are no mappings from a set S onto S*, where S* is the power set of S. The attempt at a solution This begs proof by contradiction: Let f be a mapping from S onto S*. Then for every A in S*, there is an a in S with f(a) = A. I don't know how to proceed...
  39. Shaun Culver

    Is there a notion similar to a power set for permuted and ordered elements?

    I would like if there is a notion similar to that of a "power set" where the order of the elements in a set is accounted for - the elements are permuted, and each arrangement is considered to be a separate set. For example: For three singletons: {X},{Y}, & {Z} in a set S, the "ordered &...
  40. I

    Proving Power Set Inclusion: A Simple Proof for A⊆B and P(A)⊆P(B)

    This seems like a simple proof but I'm not familiar with power set proofs If A\subseteqB then P(A) \subseteq P(B)
  41. F

    Proving Power Set of S is Equal to 2^n

    Homework Statement Hey I need to show that the power set of S is equal to 2^n where n is the number of elements (inlcuding the empty set and S). I found this kinda hard as this is just what it is defined as and not much room to work with, well i couldn't find any. Homework Equations...
  42. D

    Is the Axiom of the Power Set Redundant in Set Theory?

    Hey everyone, I am currently trying to learn a bit of set theory from Halmos' book "Naive Set Theory" since I have recently been concerned with the general notion of existence in various fields of mathematics. Now, I am reading the "axiom of the power set" and I do find it a little...
  43. O

    Is the Power Set of a Finite Set Also Finite?

    How can we prove that the power set of a finite set is finite? Thanks.
  44. JasonJo

    How Do You Prove Power Set Equations for Sets A and B?

    How in the heck do i prove these: Prove whether the following equations are true for all sets. For each one that's not always true, try to prove that one side is a subset of the other, and give a counterexample to the other direction. If neither side must be a subset of the other, give a...
  45. T

    A, B are countable. P(s), power set, and f: A-> not always countable?

    Suppose A and B are countable. Explain why P(s), power set, and f: A ->B are not necessarily countable. P(s) is only countable if A and B are finite, am I am correct? Otherwise, the power set of an infinite set is not countable. As for f: A -> B, doesn't f have to be a bijection?
  46. D

    Can An Infinite Power Set Be Defined and Is it a Set?

    A buddy and I were wondering if there is a way to define a sort of infinite power set in the following way: You can make an inclusion map from X to P(X) if map each element of X to the singleton set containing it in P(X). Thus you have this chain of maps X\subset \mathcal{P}(X) \subset...
  47. Y

    Proving Power Set Inclusion When B is Included in C

    How do you show that if B is included in C, then the power set of B is included in the power set of C?