GATE CSE 2007 | Mathematical Logic Question 35 | Discrete Mathematics

GATE CSE 2007

MCQ (Single Correct Answer)

-0.6

Which one of these first-order logic formulae is valid?

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

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

$$\exists x\left( {P\left( x \right) \wedge Q\left( x \right)} \right) \Leftrightarrow \left( {\left( {\exists xP\left( x \right)} \right) \wedge \left( {\exists xQ\left( x \right)} \right)} \right)$$

$$\forall x\exists yP\left( {x,y} \right) \Rightarrow \exists y\forall xP\left( {x,y} \right)$$

GATE CSE 2006

MCQ (Single Correct Answer)

-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.

$$\forall x[(tiger(x) \wedge lion(x)) \to $$$$\{ (hungry(x) \vee threatened(x)) \to attacks(x)\} ]$$

$$\forall x[(tiger(x) \vee lion(x)) \to $$$$\{ (hungry(x) \wedge threatened(x)) \to attacks(x)\} ]$$

$$\forall x[(tiger(x) \vee lion(x)) \to $$$$\{ attacks(x) \to (hungry(x)) \vee threatened(x))\} ]$$

$$\forall x[(tiger(x) \vee lion(x)) \to $$$$\{ (hungry(x) \vee threatened(x)) \to attacks(x)\} ]$$

GATE CSE 2006

MCQ (Single Correct Answer)

-0.6

Let $$P, Q$$, and $$R$$ be sets. Let $$\Delta $$ denote the symmetric difference operator defined as $$P\Delta Q = \left( {P \cup Q} \right) - \left( {P \cap Q} \right)$$. Using venn diagrams, determine which of the following is/are TRUE.

($${\rm I}$$) $$P\Delta \left( {Q \cap R} \right) = \left( {P\Delta Q} \right) \cap \left( {P\Delta R} \right)$$
($${\rm I}{\rm I}$$) $$P \cap \left( {Q\Delta R} \right) = \left( {P \cap Q} \right)\Delta \left( {P \cap R} \right)$$

$${\rm I}$$ only

$${\rm I}$$$${\rm I}$$ only

Neither $${\rm I}$$ nor $${\rm I}$$$${\rm I}$$

Both $${\rm I}$$ and $${\rm I}$$$${\rm I}$$

GATE CSE 2006

MCQ (Single Correct Answer)

-0.6

Consider the following first order logic formula in which $$R$$ is a binary relation symbol.
$$\forall x\forall y\left( {R\left( {x,\,y} \right) \Rightarrow R\left( {y,x} \right)} \right).$$

The formula is

Satisfiable and valid

Satisfiable and so is its negation

Unsatisfiable but its negation is valid

Satisfiable but its negation is unsatisfiable

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