The given set together with the given operations is not a vector space. list the properties of the definition that fail to hold. (select all that apply.) the set of all ordered pairs of real numbers with the operations (x, y) ⊕ (x', y') = (x + x', y + y') and r (x, y) = (x, ry).

Let us look at all the axioms of a vector space and see which axioms are broken. 
1) Associativity of addition


It's trivial to see this one holds.
2)Commutativity of addition

This one holds.
3)Identity element of addition
This means that we have zero vektor. It is pretty obvious we do, it is (0,0). 

4)Inverse elements of addition
This means that for each element in V there exists element -v such that v+(-v)=0. 
We do have inverse elements.
 
5)Compatibility of scalar multiplication with field multiplication
This one holds.

6)Identity element of scalar multiplication
Identity element of scalar multiplication is simply 1.

7)Distributivity of scalar multiplication with respect to vector addition. Let's look at the definition.

Now let's look at the example:


This one hold too.
8)Distributivity of scalar multiplication with respect to field addition
Definition of this one is: 

Let's take a look at the example:


So this one doesn't hold.
The final answer would be distributivity of scalar multiplication with respect to field addition.
Please note vectors, in this case, are (a,b) and that I did not use the dot to indicate scalar multiplication.

The 40 would include the shirt and on and on