# Examples in lin.alg. where closure/addition axioms don't hold

1. Sep 24, 2014

### gummz

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

Find examples of subsets in a coordinate space where:

(a) closure addition axiom doesn't hold but closure multiplication does hold,

(b) closure addition axiom does hold but clouser multiplication doesn't hold,

(c) where neither hold.

2. Relevant equations

None in particular, but what he means by closure addition and multiplication axioms is that you can add two elements of the subset together and that will still be an element of the subset. For multiplication, you're supposed to be able to multiply the element with any number in R, and it the product will also belong to the subset.

3. The attempt at a solution

I'm thinking complex numbers have something to do with this. (c) I think could be {-3, -2, 1}, (b) is something like {-3, -2, -1}, but it's (a) that I'm worried about. I simply have no idea for (a).

edit, scratch that I'm being silly about (b)... it's very wrong.

2. Sep 24, 2014

### Staff: Mentor

You'll need larger sets.
You can use complex numbers, but be careful if you use them as vector space over the real numbers or as vector space over complex numbers (this influences the options you have for multiplication, and also the dimension of the vector space).

A good way to start for (a) and (b): take an arbitrary element (like "1"), figure out what else has to be in your set to satisfy one of the closure axioms. Then see if the other one is satisfied, if yes you'll have to add another element to your set (because removing one does not work) and repeat the step.