The existential quantifier ∃x denotes 'there exists at least one x'. An existential statement ∃x P(x) is true if and only if P(x) is true for at least one element x in the domain. Existence proofs establish that objects with certain properties actually exist.
Translate statements like 'there is a prime number greater than 100' into symbolic form. Understand that proving existence requires producing (or inferring) at least one example.
From your study of predicates and quantified statements, you know that a predicate P(x) is an open sentence whose truth value depends on the variable x, and that a universal statement ∀x P(x) claims P holds for every element of the domain. The existential quantifier ∃ makes a weaker, one-sided claim: ∃x P(x) asserts that P(x) is true for at least one x in the domain. It doesn't say which x, or how many — just that at least one exists.
The translation between symbols and natural language is the first skill to master. "There is a prime number between 10 and 20" becomes ∃x (10 < x < 20 ∧ x is prime). Notice that x is not free — the quantifier binds it. The sentence is either true or false as a complete claim, not true-for-some-x-and-false-for-others. Existential statements live at the level of whole propositions, not open sentences. A common stumbling block is forgetting to specify the domain: ∃x (x² = 2) is false over the rationals and true over the reals, so domain matters.
Proving an existential statement requires demonstrating that at least one witness exists. The most direct strategy is a constructive proof: exhibit a specific value of x and verify P(x). To prove "there exists an even prime," you point to 2 and check it. But not all existence proofs work this way. A nonconstructive proof establishes existence without identifying which element works — for example, using the intermediate value theorem to prove a continuous function has a zero without pinning down where. Both methods are valid; which is better depends on whether an explicit witness is needed or can even be found.
Understanding ∃ also sharpens your reading of universal statements. ∀x P(x) is stronger than ∃x P(x): if P holds for all x, it certainly holds for some x. And the negation of an existential statement is universal: ¬(∃x P(x)) ≡ ∀x ¬P(x). This equivalence — that "there is no x with property P" is the same as "every x lacks property P" — is the logical backbone of proofs by contradiction and will become essential when you study how to negate complex quantified statements systematically.