The question: Let $F(x,y) = (xy) - |x||y|$ for all real numbers $x$ and $y$. $F$ is binary operation on real numbers. What is the restriction of $F$ to the set of all nonnegative real numbers?
My atempt:
Let $A$ be set of all nonnegative real numbers, then the restriction of $F$ to $A \times A$, namely $F_{|A}$ has $A \times A$ as domain, but since $F$ is defined by $(xy)-|x||y|$, we have that $(\forall x)(\forall y)([x \in A \land y \in A] \Rightarrow F(x,y) = 0)$. Thus the unique element in $F_{|A}(A \times A)$ is $0$, and therefore $F_{|A}:A \times A \rightarrow A$ is the constant function $F_{|A}(x,y) = 0$
It is correct?
Yes, because if $x>0 \rightarrow |x| = x$, so $$F(x,y) = 0, \\ \forall x,y \in A\times A.$$