Q41GATE 2007MCQ

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"?

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📖 Explanation

The statement "Not every graph is connected" means that it is not the case that all graphs are connected. This is equivalent to saying "There exists at least one graph that is not connected."
We can formalize this as:
$\exists x (Graph(x) \land \neg Connected(x))$

Let's evaluate each option:

A. $\neg \forall x (Graph(x)\Rightarrow Connected(x))$
Using the rule $\neg \forall y P(y) \equiv \exists y \neg P(y)$ and $\neg (P \Rightarrow Q) \equiv P \land \neg Q$ :
$\neg \forall x (Graph(x)\Rightarrow Connected(x)) \equiv \exists x \neg (Graph(x)\Rightarrow Connected(x)) \equiv \exists x (Graph(x) \land \neg Connected(x))$
This option correctly represents the statement.

C. $\neg \forall x(\neg Graph(x) \vee Connected(x))$
Using the rule $P \Rightarrow Q \equiv \neg P \vee Q$ :
$\neg \forall x(\neg Graph(x) \vee Connected(x)) \equiv \neg \forall x (Graph(x)\Rightarrow Connected(x))$
This is identical to option A, so it also correctly represents the statement.

B. $\exists x(Graph(x) \Rightarrow \neg Connected(x))$
In standard first-order logic, $\exists x(P(x) \Rightarrow Q(x))$ is not equivalent to $\exists x(P(x) \land Q(x))$ . However, in many contexts, especially when translating natural language where the scope of the quantifier is implicitly restricted to the predicate, this form is sometimes used heuristically. If we consider the quantifier to implicitly range over only those $x$ for which $Graph(x)$ is true, then $Graph(x)$ becomes a tautology within that scope, simplifying $Graph(x) \Rightarrow \neg Connected(x)$ to $\neg Connected(x)$ . In this specific (non-standard but common in such questions) interpretation, this option would be equivalent to $\exists x (Graph(x) \land \neg Connected(x))$ . We must assume this interpretation for B to align with the target answer.

D. $\forall x(Graph(x) \Rightarrow \neg Connected(x))$
This statement translates to "For every $x$ , if $x$ is a graph, then $x$ is not connected." In simpler terms, "Every graph is not connected" or "All graphs are disconnected." This is a much stronger statement than "Not every graph is connected" (which means "at least one graph is not connected"). If all graphs are disconnected, then it implies that not every graph is connected. However, if not every graph is connected (e.g., some are connected and some are not), it does not imply that all graphs are disconnected. Therefore, this option does not represent the statement. The change from an existential quantifier in the original statement to a universal quantifier here fundamentally alters the meaning.

Given that only one option should not represent the statement, and D represents a fundamentally different and stronger assertion due to the universal quantifier, D is the sentence that DOES NOT represent the given statement.

The final answer is $\boxed{D}$

Q42GATE 2006MCQ

Consider the following first order logic formula in which R is a binary relation symbol. $\forall x \forall y(R(x, y) \Longrightarrow R(y, x))$ , The formula is

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Q43GATE 2006MCQ

Which one of the first order predicate calculus statements given below correctly expresses the following English statement? Tigers and lions attack if they are hungry or threatened.

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Q44GATE 2006MCQ

Consider the following propositional statements: $P1:((A\wedge B)\rightarrow C)\equiv (A\rightarrow C)\wedge (B\rightarrow C)$ $P2:((A\vee B)\rightarrow C)\equiv (A\rightarrow C)\vee (B\rightarrow C)$ Which one of the following is true?

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Q45GATE 2005MCQ

What is the first order predicate calculus statement equivalent to the following? Every teacher is liked by some student

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Q46GATE 2005MCQ

Let P, Q and R be tree atomic prepositional assertions. Let X denote (P $\vee$ Q) $\rightarrow$ R and Y denote (P $\rightarrow$ R) $\vee$ (Q $\rightarrow$ R). Which one of the following is a tautology?

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Q47GATE 2005MCQ

Let P(x) and Q(x) be arbitrary predicates. Which of the following statements is always TRUE?

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Q48GATE 2004MCQ

Let p,q,r and s be four primitive statements. Consider the following arguments:

\begin{array}{l} P:[(\neg p \vee q) \wedge(r \rightarrow s) \wedge(p \vee r)] \rightarrow(\neg s \rightarrow q) \\ Q:[(\neg p \wedge q) \wedge[q \rightarrow(p \rightarrow r)]] \rightarrow \neg r \\ R:[[(q \wedge r) \rightarrow p] \wedge(\neg q \vee p)] \rightarrow r \\ S:[p \wedge(p \rightarrow r) \wedge(q \vee \neg r)] \rightarrow q \end{array}

Which of the above arguments are valid?

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Q49GATE 2004MCQ

Let a(x, y), b(x, y,) and c(x, y) be three statements with variables x and y chosen from some universe. Consider the following statement: $\qquad(\exists x)(\forall y)[(a(x, y) \wedge b(x, y)) \wedge \neg c(x, y)]$ Which one of the following is its equivalent?

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Q50GATE 2004MCQ

The following propositional statement is $(P\rightarrow (Q\vee R))\rightarrow ((P\vee Q)\rightarrow R)$

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Q51GATE 2003MCQ

Consider the following logic program P $A(x)\leftarrow B(x,y),C(y) \leftarrow B(x,x)$ Which of the following first order sentences is equivalent to P?

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Q52GATE 2003MCQ

Let (S, $\leq$ ) be a partial order with two minimal elements a and b, and a maximum element c. Let P:S $\rightarrow$ {True, False} be a predicate defined on S . Suppose that P(a) = True, P(b) =False and P(x) $\Rightarrow$ p(y) for all x,y $\in$ S satisfying x $\leq$ y, where $\Rightarrow$ stands for logical implication. Which of the following statements CANNOT be true?

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Q53GATE 2003MCQ

Which of the following is a valid first order formula? (Here $\alpha$ and $\beta$ are first order formulae with x as their only free variable)

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Q54GATE 2003MCQ

Consider the following formula $\alpha$ and its two interpretations $I_{1} \; and \; I_{2}$ $\alpha :(\forall x)[P_{x}\leftrightarrow (\forall y)[Q_{xy}\leftrightarrow \neg Q_{yy}]]\rightarrow (\forall x)[\neg P_{x}]$ I1: Domain : the set of natural numbers $P_{x}\equiv x$ is a prime number $Q_{xy}\equiv y$ divides x I2; same as I1 except that $P_{x}\equiv x$ is a composite number. Which of the following statements is true?

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Q55GATE 2003MCQ

the following resolution rule is used in logic programming: Derive clause(P $\vee$ Q) from clauses (P $\vee$ R),(Q $\vee \neg$ JR) Which of the following statements related to this rule is FASLE?

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Q56GATE 2002MCQ

"If X then Y unless Z" is represented by which of the following formulas in prepositional logic? ( $"\neg"$ , is negation, $"\wedge "$ is conjunction, and $"\rightarrow "$ is implication)

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Q57GATE 2001MCQ

Consider two well-formed formulas in prepositional logic. $F1: \; P\Rightarrow \neg P$ $F2: \;( P\Rightarrow \neg P)\vee ( \neg P\Rightarrow P)$ Which of the following statements is correct?

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Q58MCQ

Let a, b, c, d be propositions. Assume that the equivalence $a \Leftrightarrow ( b \vee \neg b) \text{ and }b \Leftrightarrow c$ hold. Then the truth-value of the formula $(a \land b) \rightarrow (a \land c) \lor d$ is always

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Q59MCQ

What is the converse of the following assertion? I stay only if you go

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Q60MCQ

Which of the following propositions is a tautology?

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