Urgent: Tell if this is Reflexive, transitive, symmetric?

1. The problem statement, all variables and given/known data

Determine if the following relation is reflexive, transitive, symmetric or anti-symmetric.
(A,B) element of R(relation) if for every epsilon > 0, there exists a element of A and b element of B with |a-b| < epsilon.

2. Relevant equations

3. The attempt at a solution
I already proved that this is a reflexive relation (please correct me if I'm wrong):
Let (A,B) be in R.
Prove that (A,A) is also in R.
NTS: For all a element of A and b element of A, |a-b| < epsilon ; epsilon>0
Proof:
Let a be element of A and b element of A (also).
|a E A - b E A| ?< epsilon
for simplicity we can write it: |a-a| < epsilon, which is true for all a and b because there's a chance that a and b will be equal since they're taken in the same set. We are sure that 0 < epsilon because epsilon > 0 by our assumption.

Now, how can I show that this is also transitive? and symmetric or antisymmetric?
Minor question, do I need to confirm that sets A and B are mutually exclusive to each other?
Thanks.

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 Quote by DevNeil 1. The problem statement, all variables and given/known data Determine if the following relation is reflexive, transitive, symmetric or anti-symmetric. (A,B) element of R(relation) if for every epsilon > 0, there exists a element of A and b element of B with |a-b| < epsilon.
You haven't told us what A and B are. One might guess they are (non-empty?) subsets of the real numbers. Is that right?

 3. The attempt at a solution I already proved that this is a reflexive relation (please correct me if I'm wrong): Let (A,B) be in R. Prove that (A,A) is also in R. NTS: For all a element of A and b element of A, |a-b| < epsilon ; epsilon>0
To prove R is reflexive you need to show (A,A) in R for all A. You don't start with "Let (A,B) in R".
 Proof: Let a be element of A and b element of A (also). |a E A - b E A| ?< epsilon for simplicity we can write it: |a-a| < epsilon, which is true for all a and b because there's a chance that a and b will be equal since they're taken in the same set. We are sure that 0 < epsilon because epsilon > 0 by our assumption.
That is a very confused paragraph. All you need to write is: Suppose ##\epsilon > 0##. Pick any a in A (A is non-empty?). ##|a - a|=0 < \epsilon## so (A,A) is in R.
 Now, how can I show that this is also transitive? and symmetric or antisymmetric?
Write down carefully what you need to prove as a first step. If it seems reasonable, try to prove it, otherwise see if you can make a counterexample.
 Minor question, do I need to confirm that sets A and B are mutually exclusive to each other? Thanks.
Who knows? Like I said above, you haven't even told us what A and B are.

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