GATE CSE 2005 | Mathematical Logic Question 40 | Discrete Mathematics

GATE CSE 2005

MCQ (Single Correct Answer)

-0.6

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

$$\left( {\forall x\left( {P\left( x \right) \vee Q\left( x \right)} \right)} \right) \Rightarrow \left( {\left( {\forall xP\left( x \right)} \right) \vee \left( {\forall xQ\left( x \right)} \right)} \right)$$

$$\left( {\forall x\left( {P\left( x \right) \Rightarrow Q\left( x \right)} \right)} \right) \Rightarrow \left( {\left( {\forall xP\left( x \right)} \right) \Rightarrow \left( {\forall xQ\left( x \right)} \right)} \right)$$

$$\left( {\left( {\forall x\left( {P\left( x \right)} \right) \Rightarrow \left( {\forall xQ\left( x \right)} \right)} \right) \Rightarrow \left( {\forall x\left( {P\left( x \right) \Rightarrow Q\left( x \right)} \right)} \right)} \right)$$

$$\left( {\left( {\forall x\left( {P\left( x \right)} \right)} \right)} \right) \Leftrightarrow \left( {\forall x\left( {Q\left( x \right)} \right)} \right) \Rightarrow \left( {\forall x\left( {P\left( x \right) \Leftrightarrow Q\left( x \right)} \right)} \right)$$

GATE CSE 2005

MCQ (Single Correct Answer)

-0.6

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

$$\forall \left( x \right)\left[ {teacher\left( x \right) \to \exists \left( y \right)\left[ {student\left( y \right) \to likes\left( {y,\,x} \right)} \right]} \right]$$

$$\forall \left( x \right)\left[ {teacher\left( x \right) \to \exists \left( y \right)\left[ {student\left( y \right) \wedge likes\left( {y,\,x} \right)} \right]} \right]$$

$$\exists \left( y \right)\forall \left( x \right)\left[ {teacher\left( x \right) \to \left[ {student\left( y \right) \wedge likes\left( {y,x} \right)} \right]} \right]$$

$$\forall \left( x \right)\left[ {teacher\left( x \right) \wedge \exists \left( y \right)\left[ {student\left( y \right) \to likes\left( {y,\,x} \right)} \right]} \right]$$

GATE CSE 2005

MCQ (Single Correct Answer)

-0.6

Let $$P, Q$$ and $$R$$ be three atomic prepositional assertions. Let $$X$$ denotes $$\left( {P \vee Q} \right) \to R$$ and $$Y$$ denote $$\left( {P \to R} \right) \vee \left( {Q \to R} \right)$$.

Which one of the following is a tautology?

$$X \equiv Y$$

$$X \to Y$$

$$Y \to X$$

$$\neg Y \to X$$

GATE CSE 2004

MCQ (Single Correct Answer)

-0.6

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

$$P:\left[ {\left( {\neg p \vee q} \right) \wedge \left( {r \to s} \right) \wedge \left( {p \vee r} \right)} \right] \to \left( {\neg s \to q} \right)$$
$$Q:\left[ {\left( {\neg p \wedge q} \right) \wedge \left[ {q \to \left( {p \to r} \right)} \right]} \right] \to \neg r$$
$$R:\left[ {\left[ {\left( {q \wedge r} \right) \to p} \right] \wedge \left( {\neg q \vee p} \right)} \right] \to r$$
$$S:\left[ {p \wedge \left( {p \to r} \right) \wedge \left( {q \vee \neg r} \right)} \right] \to q$$

Which of the above arguments are valid?

$$P$$ and $$Q$$ only

$$P$$ and $$R$$ only

$$P$$ and $$S$$ only

$$P, Q, R$$ and $$S$$

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