Proving Denumerability of Sets: Induction Method | Homework Help

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Homework Help Overview

The problem involves proving by induction that if a set S, consisting of n elements none of which are in a set A, is given, then the union of A and S is denumerable. The subject area pertains to set theory and the concept of denumerability.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the definition of denumerable and its equivalence to countable. There are questions about the implications of the problem statement and the lack of information regarding set A. Some participants suggest that the problem may not be provable without additional context or definitions.

Discussion Status

The discussion is ongoing, with participants exploring definitions and questioning the completeness of the problem statement. Some guidance has been offered regarding the definitions of denumerable and countable, but there is no consensus on how to proceed due to the ambiguity surrounding set A.

Contextual Notes

There is a noted absence of information about set A, which raises concerns about the validity of the proof. Participants also highlight the importance of having a clear definition of denumerable to approach the problem effectively.

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Homework Statement


Prove by induction that if set S = {x1,x2...,xn} is a set of n elements non of which are in a set A, then AuS is denumerable


Homework Equations


im not sure what to put here


The Attempt at a Solution


what does this mean
denumerable? and how do i start this
I know that i can say AuX when x is an element not in A
but what do i do beyond this
 
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My definition is that a set is denumerable (or countable) if it's finite or can be put into 1-1 correspondence with the set of integers. What's your definition? I'm guessing A is given to be denumerable. Then if you already know AU{x} is denumerable it should be pretty easy. Think induction.
 
If the problem asks you to prove a set is denumerable, you should already have been given a definition of "denumerable". I wouldn't be surprized if the definition were in the same chapter this problem is in. And, as tiny-tim suggested, you must have been given some information about A- without that the statement you say you want to prove is NOT true.
 
ok i will look
However, nothing more is said about a
thats the whole problem
 
mathcnc said:
ok i will look
However, nothing more is said about a
thats the whole problem

If that's all that's said then you can't prove the set is denumerable. And if you haven't been given a definition of denumerable then you super can't prove it. And Halls, I'm not tiny-tim.
 
ok so u are saying denumerable and countable are the same thing?
 
That's what I would say, but check your book for fine print issues.
 
Some books use "countable" to mean "there is a one to one function from the set onto N" and "denumerable" to mean finite or countable.

If "Prove by induction that if set S = {x1,x2...,xn} is a set of n elements non of which are in a set A, then AuS is denumerable" is all that is said, then consider A= R. The statement is not true.
 
Dick said:
If that's all that's said then you can't prove the set is denumerable. And if you haven't been given a definition of denumerable then you super can't prove it. And Halls, I'm not tiny-tim.

But you look so much alike.:biggrin:

(Seriously you and tiny tim have done wonderful work here- no wonder I get the two of you confused.)
 
  • #10
HallsofIvy said:
But you look so much alike.:biggrin:

(Seriously you and tiny tim have done wonderful work here- no wonder I get the two of you confused.)

Well, since you put it so nicely, I guess I forgive you.
 

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