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Let $\text{ Graph}(x)$ be a predicate which denotes that $x$ is a graph. Let $\text{ Connected}(x)$ be a predicate which denotes that $x$ is connected. Which of the following first order logic sentences DOES NOT represent the statement:

$\qquad\text{“Not every graph is connected"}$

1. $\lnot \forall x\, \Bigl (\text{ Graph}(x) \implies \text{ Connected}(x) \Bigr )$
2. $\exists x\, \Bigl (\text{ Graph}(x) \land \lnot \text{ Connected}(x) \Bigr )$
3. $\lnot \forall x \, \Bigl ( \lnot \text{ Graph}(x) \lor \text{ Connected}(x) \Bigr )$
4. $\forall x \, \Bigl ( \text{ Graph}(x) \implies \lnot \text{ Connected}(x) \Bigr )$

Option B is correct

Option D is incorrect as it means Every graph is not connected which is different from Not every graph is connected

Not every graph is connected
It can be rewritten as:

There is a graph which is not connected

so    ∃x(graph(x) ⋀ ¬connected(x)

or another way to do it ,simply:

¬∀x(graph(x) → connected(x))

∃x¬(graph(x) → connected(x))        (negation rule)

∃x¬(¬(graph(x) ⋁ connected(x))

And finally

∃x(graph(x) ⋀ ¬connected(x))

Question is for "DOES NOT"
lol  i didnt read that,i am srry

Answer will clearly be (D) as it says "for all x if x is a graph then it will not be connected" which is  false as although there are some graphs which are not connected  but this does not mean there are no connected graphs

Option a, b, c are same and option D stated that “all graphs aren’t connected”.
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