Cardinality of sets: prove equality

  • Thread starter Thread starter Dustinsfl
  • Start date Start date
  • Tags Tags
    Cardinality Sets
Click For Summary

Homework Help Overview

The discussion revolves around the cardinality of sets, specifically addressing the equality of the cardinality of a set A, defined as the real numbers excluding zero, and the interval (0,1). Participants explore the implications of A being countable and the contradiction that arises from this assumption.

Discussion Character

  • Conceptual clarification, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants examine the assumption that A is countable and its implications, questioning the relationship between A and the interval (0,1). Some suggest that the Continuum Hypothesis may be relevant, while others propose finding a better lower bound for the cardinality of A.

Discussion Status

The discussion is active, with participants providing insights and suggestions for further exploration. There is no explicit consensus, but various lines of reasoning are being explored regarding the cardinality of the sets involved.

Contextual Notes

Participants are navigating the constraints of cardinality definitions and the implications of countability versus uncountability in the context of set theory.

Dustinsfl
Messages
2,217
Reaction score
5
[tex]A=\mathbb{R}-0[/tex]

[tex](0,1)\subseteq \mathbb{R}-0[/tex]

Assume A is countable.

Since A is countable, then [tex]A\sim\mathbb{N}[/tex].
Which follows that [tex](0,1)\sim\mathbb{N}[/tex].

However, (0,1) is uncountable so by contradiction, Card(A)=c

Correct?
 
Physics news on Phys.org


The fact that A is not countable, together with the fact it is a subset of R only proves that
[tex]\aleph_0 < \mathop{Card}(A) \leq c[/tex]​
 


So it looks like you will have to use the Continuum Hypothesis!
 


Or find a better lower bound. :wink:
 


Think about the function that maps [itex]n\mapsto n+1[/itex] for [itex]n\in \mathbb{N}[/itex] where [itex]\mathbb{N}[/itex] is regarded as a subset of [itex]\mathbb{R}[/itex].
 


Thanks for the help
 

Similar threads

Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##1 + a=s(a)=a+1## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##
  • · Replies 1 ·
Replies
1
Views
2K
Prove ##a + b = b + a## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates
  • · Replies 3 ·
Replies
3
Views
2K
Prove ##a\cdot b = b \cdot a ##using Peano postulates
  • · Replies 3 ·
Replies
3
Views
1K
Prove ##(a+b) + c = a + (b+c)## using Peano postulates
  • · Replies 9 ·
Replies
9
Views
2K
Show Cardinality of Real Numbers and Complements
  • · Replies 11 ·
Replies
11
Views
2K
Prove if ##a\cdot c = b \cdot c## then ##a = b## using Peano postulates
  • · Replies 2 ·
Replies
2
Views
1K
Prove ##(a\cdot b)\cdot c =a\cdot (b \cdot c)## using Peano postulates
  • · Replies 1 ·
Replies
1
Views
1K