The quantifier ₁ is the uniqueness quantifier, which may also be written !. The notation ₁xP (x) [or

!xP (x)] states that there exists exactly 1 value of x that satisfies P (x). For example, ₁x(1 x = 0).

Using only universal and existential quantifiers along with logical operators,

1.express ₁xP (x) (there exists exactly 1 value of x that satisfies P (x)).

2.express ₂xP (x) (there exist exactly 2 values of x that satisfy P (x)).

3.express that there is exactly 1 pair of values ( x, y) that satisfies P (x, y).