Let $F$ be a field and let $K$ be an associative unital $F-algebra$. If $A$ and $B$ are subsets of $K$, we let $A • B$ be the set of all elements of $K$ of the form $a • b$, with $a ∈ A$ and $b ∈ B$ (in particular, $∅ • B = A • ∅ = ∅$). We know that the set $V$ of all subsets of $K$ is a vector space over $GF(2)$ with $A+B=A\cup B\setminus A\cap B, 0A=∅,1A=A$. Is $V$ a $GF(2)-algebra$?
I think this is false, but I can think of a contradiction.
This is an exercise from the book: The Linear Algebra a Beginning Graduate Student Should Know by Golan.