XOR and the null set

1. Sep 26, 2007

MathInProgress

1. The problem statement, all variables and given/known data
What does it mean if A ⊕ B = ∅? Explain

2. Relevant equations
From what I know the null set is part of every subset and A XOR B would be true if and only if exactly one of A or B would be true.

3. The attempt at a solution
If someone could please help me in formulating an explantation on A XOR B = null set

2. Sep 26, 2007

HallsofIvy

Staff Emeritus
I would have thought of XOR as a logical operator than a set operator but there is no reason why it can't be: I interpret A XOR B as meaning "those elements that are in A or in B but not in both: $A\cup B- A\cap B$. More often (to me at least) called the "symmetric difference" of A and B. If A XOR B is empty, then there must NOT be in points that are in A or in B but not in their intersection. What does that imply about $A\cap B$ and so A and B themselves?

3. Sep 26, 2007

MathInProgress

This is part b of a question. I did not list down the entire question because I thought it is not important (and it may still not be), but here it is:

The symmetric difference of two sets A and B is defined as A XOR B = (A - B) U (B -A).

b. What does it mean if A XOR B = null set?

Can I say it it this way:

Let A = {1,2,3,4,5} and B = {1,2,3,4,5} then A XOR B = null set? This is because there are no elements that are exclusive to either set.