GATE CSE 2006 | Mathematical Logic Question 44 | Discrete Mathematics

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

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 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

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

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

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

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

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