Proving Denumerability of Sets: Induction Method | Homework Help

[mathcnc]

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

 

[Dick]

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.

 

[HallsofIvy]

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.

 

[mathcnc]

ok i will look
However, nothing more is said about a
thats the whole problem

 

[Dick]

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.

 

[mathcnc]

ok so u are saying denumerable and countable are the same thing?

 

[Dick]

That's what I would say, but check your book for fine print issues.

 

[HallsofIvy]

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.

 

[HallsofIvy]

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.

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

 

[Dick]

But you look so much alike.

(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.