Negation of statement and equal sign? I am trying to negate the following statement: $$(∀,∈) ( + = )$$ I turned this into: $$∃ ∈ S ¬(x, y) (x - y ≠ xy)$$ Is this correct? What is the right way to negate the plus and equal sign?

Consider the following

$$(\exists x,y\in S)(x+y\
e xy)$$

What does it mean for the above sentence to be false? Do you see the corrections you need to make?

EDIT:

Back to the original question. The negation of $$(\forall x,y\in S)(x+y= xy)$$ is indeed the following $$\
eg(\forall x,y\in S)(x+y= xy)\tag{1}$$ but you can "simplify" (that's a matter of perspective) by moving the negation past the quantifier $\forall$ to get $$(\exists x,y\in S)\
eg(x+y= xy).\tag{2}$$ Last step is to negate an equality statement. Thus we obtain $$(\exists x,y\in S)(x+y\
e xy).\tag{3}$$

I want to emphasize $(1\text{-}3)$ are all correct.