Existential Statement is a #202204281244 which states there is at least one thing for which the property is true, given that it could be either true or false. It usually uses existential quantifier \(\exists\) which express there is a, there exists, we can find a, there is at least one, for some and for at least one.
For example: There is a prime number that is even can be denoted in mathematics as \(\exists p \in P \text{ such that } p \text{ is a prime number}\) where \(P\) represents all positive integers, which read There exists a \(p\) in the set of all positive integers such that \(p\) is a prime number.
To prove this, one can just show that at least one \(p\) in the set of \(P\) is true. To prove that the statement is wrong, one could not avoid proving that all of them are false, which is called the method of exhaustion. You can see the difference between it and 202204281245.
The negation of an Existential Statement is logically equivalent# to a 202204281245. Formally, this means that \(\sim (\exists x \in D \text{ such that } Q(x)) \equiv \forall x \in D, \sim Q(x)\). For instance, the negation of the statement “Some books have a cover” is “All books don’t have a cover” or “No books have a cover”. As you have noticed, the negation of it is to disprove every element in the set has the property \(Q(x)\), which is quite similar to the reasoning of the method of exhaustion.