1
GATE CSE 2005
+2
-0.6
What is the first order predicate calculus statement equivalent to the following?
Every teacher is liked by some student
A
$$\forall \left( x \right)\left[ {teacher\left( x \right) \to \exists \left( y \right)\left[ {student\left( y \right) \to likes\left( {y,\,x} \right)} \right]} \right]$$
B
$$\forall \left( x \right)\left[ {teacher\left( x \right) \to \exists \left( y \right)\left[ {student\left( y \right) \wedge likes\left( {y,\,x} \right)} \right]} \right]$$
C
$$\exists \left( y \right)\forall \left( x \right)\left[ {teacher\left( x \right) \to \left[ {student\left( y \right) \wedge likes\left( {y,x} \right)} \right]} \right]$$
D
$$\forall \left( x \right)\left[ {teacher\left( x \right) \wedge \exists \left( y \right)\left[ {student\left( y \right) \to likes\left( {y,\,x} \right)} \right]} \right]$$
2
GATE CSE 2004
+2
-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?

A
$$P$$ and $$Q$$ only
B
$$P$$ and $$R$$ only
C
$$P$$ and $$S$$ only
D
$$P, Q, R$$ and $$S$$
3
GATE CSE 2004
+2
-0.6
The following propositional statement is $$\left( {P \to \left( {Q \vee R} \right)} \right) \to \left( {\left( {P \wedge Q} \right) \to R} \right)$$\$
A
Satisfiable but not valid
B
Valid
C
D
None of the above
4
GATE CSE 2003
+2
-0.6
The following resolution rule is used in logic programming. Derive clause $$\left( {P \vee Q} \right)$$ from clauses $$\left( {P \vee R} \right)$$, $$\left( {Q \vee \neg R} \right)$$.

Which of the following statements related to this rule is FALSE?

A
$$\left( {\left( {P \vee R} \right) \wedge \left( {Q \vee \neg R} \right)} \right) \Rightarrow \left( {P \vee Q} \right)$$ is logically valid
B
$$\left( {P \vee Q} \right) \Rightarrow \left( {\left( {P \vee R} \right) \wedge \left( {Q \vee \neg R} \right)} \right)$$ is logically valid
C
$$\left( {P \vee Q} \right)$$ is satisfiable if and only if $${\left( {P \vee R} \right) \wedge \left( {Q \vee \neg R} \right)}$$ is satisfiable
D
$$\left( {P \vee Q} \right) \Rightarrow$$ FALSE if and only if both $$P$$ and $$Q$$ are unsatisfiable
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