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),
P
{\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.
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...
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...
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...
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...
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...
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...
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
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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).
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...
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 ?
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}}?
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...
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...
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.
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...
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...
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...
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...
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...
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 =...
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...
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...
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...
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...
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 &...
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...
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...
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...
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?
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...