Can infinite sets differ finitely?

[Loren Booda]

Can two infinite sets differ by a finite number of elements?

 

[Moo Of Doom]

Take for example A = \mathbb{N} and B = \mathbb{N} \setminus \left\{ 1 \right\}. Then A and B are both infinite sets, and they share all elements except for 1. Or is this not what you mean by "differ"?

 

[CRGreathouse]

Can two infinite sets differ by a finite number of elements?

I take your question as:

"Given two infinite sets A and B, is the symmetric difference (A\cup B)\setminus(A\cap B) ever nonempty?"

in which case the above post gives an example of when this can occur.

 

[Loren Booda]

Please forgive my ignorance, but what does \ mean in this context?

 

[Dragonfall]

Let' say A = {0, 1, 2, ...} and B = {1, 2, 3, ...}. A and B are both infinite, but A and B "differ" by one element. Namely 0.

 

[CRGreathouse]

Please forgive my ignorance, but what does \ mean in this context?

Set subtraction (thus the LaTeX command \setminus). {1, 2, 3} \ {2} = {1, 3}.

 

[HallsofIvy]

I take your question as:

"Given two infinite sets A and B, is the symmetric difference (A\cup B)\setminus(A\cap B) ever nonempty?"

in which case the above post gives an example of when this can occur.

Please forgive my ignorance, but what does \ mean in this context?

It's the "set difference". A\B is "All values that are in A but not in B". Think "A take away any members of A intersect B".

 

[Loren Booda]

Thanks, all.

 

[Loren Booda]

There isn't an inverse function to \, is there?

 

[Dragonfall]

Yes there is, the union. A \ B U B = A.

 

[secretman]

Not quite; only if B is contained in A.

 

[HallsofIvy]

Not quite; only if B is contained in A.

Only if B is a subset of A.

Using "is contained in" to mean "is a subset of" can cause an awful lot of confusion.