1
GATE CSE 2008
+2
-0.6
If $$P$$, $$Q$$, $$R$$ are Boolean variables, then $$(P + \bar{Q}) (P.\bar{Q} + P.R) (\bar{P}.\bar{R} + \bar{Q})$$ simplifies to
A
$$P.\overline Q$$
B
$$P.\overline R$$
C
$$P.\overline Q + R$$
D
$$P.\overline R + Q$$
2
GATE CSE 2008
+2
-0.6
Let fsa and $$pda$$ be two predicates such that fsa$$(x)$$ means $$x$$ is a finite state automation, and pda$$(y)$$ means that $$y$$ is a pushdown automation. Let $$equivalent$$ be another predicate such that $$equivalent$$$$(a,b)$$ means $$a$$ and $$b$$ are equivalent. Which of the following first order logic statements represents the following:

Each finite state automation has an equivalent pushdown automation.

A
$$\left( {\forall x\,\,fsa\left( x \right)} \right) \Rightarrow \left( {\exists y\,\,pda\left( y \right) \wedge \,equivalent\,\,\left( {x,\,y} \right)} \right)$$
B
$$\sim \forall y\left( {\exists x\,\,fsa\left( x \right) \Rightarrow pda\left( y \right) \wedge \,equivalent\left( {x,\,y} \right)} \right)$$
C
$$\forall x\,\exists y\left( {fsa\left( x \right) \wedge pda\left( y \right) \wedge \,equivalent\left( {x,\,y} \right)} \right)$$
D
$$\forall x\,\exists y\left( {fsa\left( y \right) \wedge pda\left( x \right) \wedge \,equivalent\left( {x,\,y} \right)} \right)$$
3
GATE CSE 2008
+2
-0.6
Which of the following first order formulae is logically valid? Here $$\alpha \left( x \right)$$ is a first order formulae with $$x$$ as a free variable, and $$\beta$$ is a first order formula with no free variable.
A
$$\left[ {\beta \to \left( {\exists x,\alpha \left( x \right)} \right)} \right] \to \left[ {\forall x,\beta \to \alpha \left( x \right)} \right]$$
B
$$\left[ {\exists x,\beta \to \alpha \left( x \right)} \right] \to \left[ {\beta \to \left( {\forall x,\alpha \left( x \right)} \right)} \right]$$
C
$$\left[ {\left( {\exists x,\alpha \left( x \right)} \right) \to \beta } \right] \to \left[ {\forall x,\alpha \left( x \right) \to \beta } \right]$$
D
$$\left[ {\left( {\forall x,\alpha \left( x \right)} \right) \to \beta } \right] \to \left[ {\forall x,\alpha \left( x \right) \to \beta } \right]$$
4
GATE CSE 2008
+2
-0.6
$$P$$ and $$Q$$ are two propositions. Which of the following logical expressions are equivalent?

$${\rm I}.$$ $${\rm P}\, \vee \sim Q$$
$${\rm I}{\rm I}.$$ $$\sim \left( { \sim {\rm P} \wedge Q} \right)$$
$${\rm I}{\rm I}{\rm I}.$$ $$\left( {{\rm P} \wedge Q} \right) \vee \left( {{\rm P} \wedge \sim Q} \right) \vee \left( { \sim {\rm P} \wedge \sim Q} \right)$$
$${\rm I}V.$$ $$\left( {{\rm P} \wedge Q} \right) \vee \left( {{\rm P} \wedge \sim Q} \right) \vee \left( { \sim {\rm P} \wedge Q} \right)$$

A
Only $${\rm I}$$ and $${\rm I}$$$${\rm I}$$
B
Only $${\rm I}$$, $${\rm I}$$$${\rm I}$$ and $${\rm I}$$$${\rm I}$$$${\rm I}$$
C
Only $${\rm I}$$, $${\rm I}$$$${\rm I}$$ and $${\rm I}$$$$V$$
D
All of $${\rm I}$$, $${\rm I}$$$${\rm I}$$, $${\rm I}$$$${\rm I}$$$${\rm I}$$ and $${\rm I}$$$$V$$
GATE CSE Subjects
Discrete Mathematics
Programming Languages
Theory of Computation
Operating Systems
Digital Logic
Computer Organization
Database Management System
Data Structures
Computer Networks
Algorithms
Compiler Design
Software Engineering
Web Technologies
General Aptitude
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