Disproving a Claim

[TheWrathOfMath]

Prove or disprove the following claim:
V is a vector space; U and W are vector subspaces of V.
Let S be the group of all vectors is V which do not exist in the intersection of U and W, in addition to the zero vector.
S is a vector space (over the same field, and with the same operations as in V).

I need to find a counterexample.

Does this work?

V=R3
U=(x,y,0)
W=(0,y,z)
U∩W=(0,y,0)

S={v∈R^3| v∉(0,y,0)}U{0}

I found the following example:

u=(2, -8, 2)∈S, v=(-2, 9, -2)∈S
(because they are not on the y-axis).

u+v=(0,1,0)∉ S
(because it is on the y-axis).

This counterexample works, right?

 

[blamocur]

Prove or disprove the following claim:
V is a vector space; U and W are vector subspaces of V.
Let S be the group of all vectors is V which do not exist in the intersection of U and W, in addition to the zero vector.
S is a vector space (over the same field, and with the same operations as in V).

I need to find a counterexample.

Does this work?

V=R3
U=(x,y,0)
W=(0,y,z)
U∩W=(0,y,0)

S={v∈R^3| v∉(0,y,0)}U{0}

I found the following example:

u=(2, -8, 2)∈S, v=(-2, 9, -2)∈S
(because they are not on the y-axis).

u+v=(0,1,0)∉ S
(because it is on the y-axis).

This counterexample works, right?

Works for me.