1
GATE CSE 2007
+2
-0.6
Which one of these first-order logic formulae is valid?
A
$$\forall x\left( {P\left( x \right) \Rightarrow Q\left( x \right)} \right) \Rightarrow \left( {\left( {\forall xP\left( x \right)} \right) \Rightarrow \left( {\forall xQ\left( x \right)} \right)} \right)$$
B
$$\exists x\left( {P\left( x \right) \vee Q\left( x \right)} \right) \Rightarrow \left( {\left( {\exists xP\left( x \right)} \right) \Rightarrow \left( {\exists xQ\left( x \right)} \right)} \right)$$
C
$$\exists x\left( {P\left( x \right) \wedge Q\left( x \right)} \right) \Leftrightarrow \left( {\left( {\exists xP\left( x \right)} \right) \wedge \left( {\exists xQ\left( x \right)} \right)} \right)$$
D
$$\forall x\exists yP\left( {x,y} \right) \Rightarrow \exists y\forall xP\left( {x,y} \right)$$
2
GATE CSE 2006
+2
-0.6
Which one of the first order predicate calculus statements given below correctly expresses the following English statement?

Tigers and lion attack if they are hungry of threatened.

A
$$\forall x[(tiger(x) \wedge lion(x)) \to$$$$\{ (hungry(x) \vee threatened(x)) \to attacks(x)\} ]$$
B
$$\forall x[(tiger(x) \vee lion(x)) \to$$$$\{ (hungry(x) \wedge threatened(x)) \to attacks(x)\} ]$$
C
$$\forall x[(tiger(x) \vee lion(x)) \to$$$$\{ attacks(x) \to (hungry(x)) \vee threatened(x))\} ]$$
D
$$\forall x[(tiger(x) \vee lion(x)) \to$$$$\{ (hungry(x) \vee threatened(x)) \to attacks(x)\} ]$$
3
GATE CSE 2006
+2
-0.6
Consider the following propositional statements:

$${\rm P}1:\,\,\left( {\left( {A \wedge B} \right) \to C} \right) \equiv \left( {\left( {A \to C} \right) \wedge \left( {B \to C} \right)} \right)$$
$${\rm P}2:\,\,\left( {\left( {A \vee B} \right) \to C} \right) \equiv \left( {\left( {A \to C} \right) \vee \left( {B \to C} \right)} \right)$$ Which one of the following is true?

A
$$P1$$ is tautology, but not $$P2$$
B
$$P2$$ is tautology, but not $$P1$$
C
$$P1$$ and $$P2$$ are both tautologies
D
Both $$P1$$ and $$P2$$ are not tautologies
4
GATE CSE 2006
+2
-0.6
A logical binary relation $$\odot$$, is defined as follows:

Let ~ be the unary negation (NOT) operator, with higher precedence then $$\odot$$. Which one of the following is equivalent to $$A \wedge B?$$

A
$$\left( { \sim A \odot B} \right)$$
B
$$\left( { \sim A \odot \sim B} \right)$$
C
$$\sim \left( { \sim A \odot \sim B} \right)$$
D
$$\sim \left( { \sim A \odot B} \right)$$
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