2017 November NTA UGC NET Paper 1
October 24, 2023GATE 2007
October 24, 2023GATE 2007
Question 22

Let Graph(x) be a predicate which denotes that x is a graph. Let Connected(x) be a predicate which denotes that x is connected. Which of the following first order logic sentences DOES NOT represent the statement: “Not every graph is connected”?
¬∀x (Graph (x) ⇒ Connected (x))


¬∃x (Graph (x) ∧ ¬Connected (x))


¬∀x (¬Graph (x) ∨ Connected (x))


∀x (Graph (x) ⇒ ¬Connected (x))

Question 22 Explanation:
Option (A) and (C) are same, because in option (C) the given expression is
Given expression is
¬∀x(¬Graph(x) ∨ Connected(x)
which can be rewritten as,
¬∀x(Graph(x) ⇒ Connected(x)
which is equivalent to option (A)
(∵ ¬p∨q ≡ p→q)
So, option (A) and (C) cannot be the answer.
Coming to option (B), the given expression is,
∃x (Graph (x) ∧ ¬Connected (x))
“There exist some graph which is not connected”, which is equivalent in saying that “Not every graph is connected”.
Coming to option (D),
For all x graph is not connected, which is not correct.
Hence, option (D) is the answer.
Given expression is
¬∀x(¬Graph(x) ∨ Connected(x)
which can be rewritten as,
¬∀x(Graph(x) ⇒ Connected(x)
which is equivalent to option (A)
(∵ ¬p∨q ≡ p→q)
So, option (A) and (C) cannot be the answer.
Coming to option (B), the given expression is,
∃x (Graph (x) ∧ ¬Connected (x))
“There exist some graph which is not connected”, which is equivalent in saying that “Not every graph is connected”.
Coming to option (D),
For all x graph is not connected, which is not correct.
Hence, option (D) is the answer.
Correct Answer: D
Question 22 Explanation:
Option (A) and (C) are same, because in option (C) the given expression is
Given expression is
¬∀x(¬Graph(x) ∨ Connected(x)
which can be rewritten as,
¬∀x(Graph(x) ⇒ Connected(x)
which is equivalent to option (A)
(∵ ¬p∨q ≡ p→q)
So, option (A) and (C) cannot be the answer.
Coming to option (B), the given expression is,
∃x (Graph (x) ∧ ¬Connected (x))
“There exist some graph which is not connected”, which is equivalent in saying that “Not every graph is connected”.
Coming to option (D),
For all x graph is not connected, which is not correct.
Hence, option (D) is the answer.
Given expression is
¬∀x(¬Graph(x) ∨ Connected(x)
which can be rewritten as,
¬∀x(Graph(x) ⇒ Connected(x)
which is equivalent to option (A)
(∵ ¬p∨q ≡ p→q)
So, option (A) and (C) cannot be the answer.
Coming to option (B), the given expression is,
∃x (Graph (x) ∧ ¬Connected (x))
“There exist some graph which is not connected”, which is equivalent in saying that “Not every graph is connected”.
Coming to option (D),
For all x graph is not connected, which is not correct.
Hence, option (D) is the answer.
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