Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x)…

Conclusion: the formula that does not represent “Not every graph is connected” is the one that asserts every graph is not connected.

Detailed reasoning:

• The statement "Not every graph is connected" is naturally expressed as an existential: there exists an x such that x is a graph and x is not connected, i.e. ∃x (Graph(x) ∧ ¬Connected(x)).

• Negating the universal form gives the equivalent formulation: ¬∀x (Graph(x) ⇒ Connected(x)). Using the equivalence (P ⇒ Q) ≡ (¬P ∨ Q), this is the same as ¬∀x (¬Graph(x) ∨ Connected(x)).

• Applying ¬∀x φ ≡ ∃x ¬φ to ¬∀x (Graph(x) ⇒ Connected(x)) yields ∃x ¬(Graph(x) ⇒ Connected(x)), and ¬(Graph(x) ⇒ Connected(x)) is equivalent to (Graph(x) ∧ ¬Connected(x)). So the existential and the negated-universal forms are all equivalent.

• In contrast, the formula ∀x (Graph(x) ⇒ ¬Connected(x)) asserts that every graph is not connected (no graph is connected). That is a stronger claim and is not equivalent to "not every graph is connected."

Therefore the correct identification is that the formula stating ∀x (Graph(x) ⇒ ¬Connected(x)) does NOT represent the statement "Not every graph is connected." The other three formulas are equivalent ways of expressing the intended statement.