Let A be the set of circles in R^2 centred at (0, 0), and let B be the set of circles in R^2 centred at (1, 1). Let P(A, B) denote the open sentence A and B have exactly two points in common on the domain A × B. Consider the quantified statement S : ∀A ∈ A, ∃B ∈ B s.t. P(A, B).
(a) State S in words. (c) State ∼ S in words.
For part (a) I said:
For all circles A in R^2 centred at (0, 0) there exists a circle B in R^2 centred at (1, 1) such that A and B have exactly two points in common.
for part (a) do I have to mention the two circles have exactly two points in common in the domain A x B?
PS: sorry for the poor formatting. I'm new to this site so I'm not too familiar with the formatting used on this site.
You should include that the two circles $A$ and $B$ have exactly two points in common that must be from $\mathbf{A}\times\mathbf{B}$. By the way, it seems weird that the points must come from $\mathbf{A}\times\mathbf{B}$, and not $\mathbb{R}^2$, but what you've wrote is correct.