Give the negation of the statement
∃! x ∈ U [P (x)].
Look closley this is so simple
To find the negation of a statement, we need to negate the entire statement as well as the quantifier.
The given statement is ∃! x ∈ U [P(x)], which can be read as "There exists a unique x in the universe U such that P(x) is true."
To negate this statement, we need to change the quantifier to its negation and negate the inner statement. The negation of the existential quantifier (∃) is the universal quantifier (∀), and the negation of the uniqueness (∃!) is the existence (∃).
Therefore, the negation of the given statement is ∀x ∈ U [¬P(x)], which can be read as "For every x in the universe U, P(x) is not true."