filmov
tv
Formula for Cardinality of Power Sets | Set Theory
Показать описание
What is the formula for the cardinality of power sets? Why does it work? We go over all of that in today's math lesson! Recall that the power set, of a set A, is the set containing all subsets of A. So, for example, let A = { 1, 2, 3 }. Then, the power set of A, denoted P(A), is
P(A) = { { }, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }.
Every subset of A is an element of P(A). So how many elements are in P(A)? If we count them up, we see there are 8 elements, which is equal to 2^3. Notice that |A| = 3. So |P(A)| = 2^(|A|). This is true in general. If the cardinality of a set S is n, then the cardinality of P(S) is 2^n. So why does this work?
We have to ask ourselves how many subsets any given set has, as this number is the same as the cardinality of the power set. If our set S has n elements, then how many subsets of S are there? Well, since S has n elements, we can write it like this:
S = { 1, 2, 3, ..., n },
where each number is just a label for each of the n elements of S. Then, if we are creating a subset of S, it could either contain or not contain the element labeled 1. So we have 2 possibilities. Then, for each of those two possibilities, our subset could either contain or not contain the element labeled 2, so now we are up to 2*2 possibilities. This reasoning continues up to the nth element, where we then have 2^n possibilities. This is where the formula comes from. There are 2^n subsets of any set with n elements, so the power set, which is the set of all subsets, has cardinality 2^n.
I hope you find this video helpful, and be sure to ask any questions down in the comments!
********************************************************************
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.
********************************************************************
+WRATH OF MATH+
Follow Wrath of Math on...
P(A) = { { }, { 1 }, { 2 }, { 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 }, { 1, 2, 3 } }.
Every subset of A is an element of P(A). So how many elements are in P(A)? If we count them up, we see there are 8 elements, which is equal to 2^3. Notice that |A| = 3. So |P(A)| = 2^(|A|). This is true in general. If the cardinality of a set S is n, then the cardinality of P(S) is 2^n. So why does this work?
We have to ask ourselves how many subsets any given set has, as this number is the same as the cardinality of the power set. If our set S has n elements, then how many subsets of S are there? Well, since S has n elements, we can write it like this:
S = { 1, 2, 3, ..., n },
where each number is just a label for each of the n elements of S. Then, if we are creating a subset of S, it could either contain or not contain the element labeled 1. So we have 2 possibilities. Then, for each of those two possibilities, our subset could either contain or not contain the element labeled 2, so now we are up to 2*2 possibilities. This reasoning continues up to the nth element, where we then have 2^n possibilities. This is where the formula comes from. There are 2^n subsets of any set with n elements, so the power set, which is the set of all subsets, has cardinality 2^n.
I hope you find this video helpful, and be sure to ask any questions down in the comments!
********************************************************************
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.
********************************************************************
+WRATH OF MATH+
Follow Wrath of Math on...
0:05:25
Formula for Cardinality of Power Sets | Set Theory
0:04:05
What is a Power Set? | Set Theory, Subsets, Cardinality
0:14:43
Power Sets and the Cardinality of the Continuum
0:22:48
Cardinality of the Continuum
0:00:21
Number of elements of the Power Set Class 8 #shorts
0:03:08
Cardinality of Sets
0:06:28
Example: Cardinality of Nested Power Sets
0:00:49
Power set #shorts #youtubeshorts
0:07:19
Finding Power Set | Power Set of Empty Set | Cardinality of power set |Sets in Maths |Math Dot Com
0:13:32
cardinality of the power set proof
0:09:15
Cardinality and Constructing Larger Infinites | Nathan Dalaklis
0:00:25
Define Cardinality of a Set #shorts
0:09:46
Proof: Number of Subsets using Induction | Set Theory
0:05:03
Cardinality of a Set | @MathTeacherGon
0:00:35
Cardinality of set #math #cardinality
0:00:27
Cardinality and power of a set #mathsatoka #maths
0:02:00
Cardinality of the Power Set
0:08:01
Cardinality of the Power Set of Finite Sets - A Classic Proof
0:10:35
Finding Power Set Examples | Set Theory, Subsets and Power Sets
0:05:01
Caridnality of the Union of Two Sets (Formula Explained) | Set Theory, Cardinality, Set Union
0:23:16
Cantor's Theorem on the cardinality of a set and of its powerset (countable and uncountable set...
0:02:03
Determine the Power Set of a Set and the Cardinality of a Power Set
0:04:33
Power Set
0:50:18
Fundamentals of Mathematics - Lecture 31: The Power Set of the Naturals, Cardinality Continuum
Комментарии