What is the closure property of addition in Boolean groups?

[Artusartos]

Homework Statement

I attached the question...

I'm having trouble proving the closure property of the addition operation.

Homework Equations

The Attempt at a Solution

Since the textbook states "The Boolean group B(X) is the family of all the subsets of X"

I need to show that if A is in B(X) and B is in B(X), then A+B is in B(X). But I'm not sure how...since both A and B are subsets of X, I need to show that A+B is a subset...but X is just arbitrary. I'm a bit confused about this...

Thanks in advance

 

[rasmhop]

You know that A and B are subsets of X. From this can you figure out how to show that:
$$(A-B) \cup (B-A)$$
is a subset of X? This is all you need to prove the closure of addition.

Maybe you are confused because earlier you learned:
$$A+B = \{a+b|a\in A,\,b \in B\}$$
But note that the + operation is redefined in this exercise to mean:
$$A+B=(A-B) \cup (B-A)$$

 

[Artusartos]

You know that A and B are subsets of X. From this can you figure out how to show that:
$$(A-B) \cup (B-A)$$
is a subset of X? This is all you need to prove the closure of addition.

Maybe you are confused because earlier you learned:
$$A+B = \{a+b|a\in A,\,b \in B\}$$
But note that the + operation is redefined in this exercise to mean:
$$A+B=(A-B) \cup (B-A)$$

Thanks, but the reason that I'm confused is...isn't it already obvious that $$(A-B) \cup (B-A)$$ is a set? So how can I "prove" it?

 

[rasmhop]

Thanks, but the reason that I'm confused is...isn't it already obvious that $$(A-B) \cup (B-A)$$ is a set? So how can I "prove" it?

Yes it is, but you also need to prove that it is a subset of X. That too is fairly obvious. Just mention that A is a subset of X so A-B is a subset of X, similarily B-A is a subset of X. Therefore (A-B) union (B-A) is a subset of X. If this is a slightly more advanced course you may even just say "Obviously $(B-A)\cup (A-B)$ is a subset of X".

However you need to check that the operations are associative, that multiplication distributes over addition, and you must find additive and multiplicative identities. Furthermore you must show that every subset of X has a negation. These parts of the exercise are non-trivial (and there are still a few parts left I didn't mention such as commutativity).