# Thread: Ring, field, Galois-Field, Vector Space

1. ## 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

2. Originally Posted by Herbststurm
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
You need to show that $\displaystyle (\mathcal{P}(X),\Delta,\cap)$ satisfy all the properties of a ring. For example, $\displaystyle \Delta$ is commutative because $\displaystyle A\Delta B = B\Delta A$. Also $\displaystyle \cap$ is commutative since $\displaystyle A\cap B = B\cap A$. You also need to show distributive laws $\displaystyle A \cap (B\Delta C) = (A\cap B)\Delta (A\cap C)$. And you need to show that there is an identity element for $\displaystyle \Delta$ and inverses for it as well.

Is P(x) with this ringstructure a field?
Is there an element $\displaystyle I$ so that $\displaystyle I\cap A = A$? Yes! Take $\displaystyle X$ then since $\displaystyle A\subseteq X$ it will follow $\displaystyle A\cap X = A$. Thus, $\displaystyle X$ is the identity element for $\displaystyle \cap$. However, does every element have an inverse i.e. for any $\displaystyle A\in X$ there is $\displaystyle B\in X$ so that $\displaystyle A\cap B = X$. And the answer is of course not.

3.) Proof that P(X) is a Vector Space over the Galois-field.
Not sure I understand what exactly you trying to show here.

3. Hi,

thanks for reply. The main problem now is that I have problems to undertand the meaning of the triple, concrete the singe symbols. Of course I know what an intersection is, but why it stands alone? On what relates this?

greetings

oh, algebra is quite difficult. Just calculate things is more easier

4. ## Re: Ring, field, Galois-Field, Vector Space

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

5. ## 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 *.