Set Theory: Proving D has 2^d Subsets of Cardinality d

Click For Summary
SUMMARY

The discussion centers on proving that a set D with cardinality d has 2^d subsets of cardinality d. The user proposes splitting D into two subsets, C1 and C2, each with cardinality d, leading to the conclusion that there are d^d subsets containing C1. The user asserts that since d is an infinite cardinal, d^d equals 2^d, establishing the desired result. The conversation highlights the importance of understanding cardinality and the implications of infinite sets in set theory.

PREREQUISITES
  • Understanding of cardinality in set theory
  • Familiarity with infinite sets and their properties
  • Knowledge of basic set operations and subsets
  • Concept of exponentiation in the context of cardinal numbers
NEXT STEPS
  • Study the properties of infinite cardinal numbers
  • Explore the concept of power sets and their cardinalities
  • Learn about Cantor's theorem and its implications for set theory
  • Investigate the relationship between cardinality and set partitions
USEFUL FOR

Mathematicians, students of set theory, and anyone interested in the foundations of mathematics and the properties of infinite sets.

Punkyc7
Messages
415
Reaction score
0
Let D be a set that has cardinality d WTS that D has 2[itex]^{d}[/itex] subsets of cardinal number d.


So I was thinking about slitting D into two sets C[itex]_{1}[/itex] and C[itex]_{2}[/itex] both of cardinality d. From there I think that there are d[itex]^{d}[/itex] subsets that contain C[itex]_{1}[/itex]. Since d is an infinite cardinal d[itex]^{d}[/itex]=2[itex]^{d}[/itex].

Does that work or am I missing something>
 
Physics news on Phys.org
I have no idea what your question is. What does 'WTS' mean? And what are you trying to do?
 
want to show what I said I want to show. Were is the confusion?
 

Similar threads

Writing some ZF axioms with FOL symbols
  • · Replies 2 ·
Replies
2
Views
2K
Determine whether ## S[y] ## has a maximum or a minimum
  • · Replies 18 ·
Replies
18
Views
3K
Equivalence Relations and Counter Examples for Equinumerous Sets
  • · Replies 4 ·
Replies
4
Views
2K
Is the set open, closed, neither, bounded, connected?
  • · Replies 7 ·
Replies
7
Views
3K
Proving differentiability for inverse function on given interval
  • · Replies 3 ·
Replies
3
Views
1K
Combinatorics: looking for an alternative solution
  • · Replies 7 ·
Replies
7
Views
2K
Proving Properties of Open Sets in $\mathbb{R}^d$
  • · Replies 5 ·
Replies
5
Views
2K
Understanding of the Metric Space axioms - (axiom 2 only)
  • · Replies 19 ·
Replies
19
Views
3K
Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?
  • · Replies 2 ·
Replies
2
Views
2K
Is There an Infinite Set Smaller Than the Natural Numbers?
  • · Replies 2 ·
Replies
2
Views
4K