Can infinite sets differ finitely?

1. Oct 3, 2008

Loren Booda

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

2. Oct 3, 2008

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"?

3. Oct 3, 2008

CRGreathouse

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.

4. Oct 3, 2008

Loren Booda

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

5. Oct 3, 2008

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.

6. Oct 3, 2008

CRGreathouse

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

7. Oct 3, 2008

HallsofIvy

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".

8. Oct 3, 2008

Thanks, all.

9. Oct 8, 2008

Loren Booda

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

10. Oct 8, 2008

Dragonfall

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

11. Oct 26, 2008

secretman

Not quite; only if B is contained in A.

12. Oct 26, 2008

HallsofIvy

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.

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