Ring, field, Galois-Field, Vector Space

Hello,

We have a not empty set X and call P(X) the power set of X. For the subsets A,B in X we define the symetric difference:

$\displaystyle A \Delta B = \left(A \setminus B \right) \cup \left(B \setminus A \right) $

To proof:

1.)

Proof that the triple $\displaystyle \mathfrak{P}(X),\Delta,\cap$ is a Ring

2.)

Is P(x) with this ringstructure a field?

3.) Proof that P(X) is a Vector Space over the Galois-field.

I have no idea :(

The first questions:

What is Delta?

Why is in the triple the intersection and not the union like in the definition?

How should I start?

Thanks for help :)

all the best

Herbststurm

Re: Ring, field, Galois-Field, Vector Space

To just calculate things is easier. (Not more easier)

Re: Ring, field, Galois-Field, Vector Space

Herbsturm:

The triple gives information on what the set is (namely, P(X)), delta is an operation, and intersection another one. The operation is to be done on elements of P(X), i.e. subsets. Compare with integers Z; (Z,+,*) is a ring, you can think of delta and intersection as + and *.