Let p(x) be some property that an input x may or may not have.
(a) explain how the following sentence says ‘there is exactly one x with the property p(x):
∃x(p(x) ∧ ∀y(p(y) → y = x))
(b) write similar sentences using quantifiers for
i. there are exactly two things with the property represented by p.
ii. there are at least two things with the property represented by p.
iii. there is at most one thing with the property represented by p.
note: the shorthand ∃! p(x) is a common shorthand for ‘there is exactly one x with the property p(x). do not use this shorthand in your solutions above!