# 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

"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

Staff Emeritus
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

Staff Emeritus
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.