1
GATE CSE 2009
+2
-0.6
The binary operation ◻ is defined as follows:

Which one of the following is equivalence to $$P \vee Q$$?

A
$$\neg \,Q$$ ◻ $$\neg \,P$$
B
$$P$$ ◻ $$\neg \,Q$$
C
$$\neg \,P$$ ◻ $$Q$$
D
$$\neg \,P$$ ◻ $$\neg \,Q$$
2
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
$$P.\overline R + Q$$
3
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
$$\forall x\,\exists y\left( {fsa\left( y \right) \wedge pda\left( x \right) \wedge \,equivalent\left( {x,\,y} \right)} \right)$$
4
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
$$\left[ {\left( {\forall x,\alpha \left( x \right)} \right) \to \beta } \right] \to \left[ {\forall x,\alpha \left( x \right) \to \beta } \right]$$
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