1
GATE CSE 2013
MCQ (More than One Correct Answer)
+2
-0.6
Which one of the following is NOT logically equivalent to $$\neg \exists x\left( {\forall y\left( \alpha \right) \wedge \left( {\forall z\left( \beta \right)} \right)} \right)?$$
A
$$\forall x\left( {\exists z\left( {\neg \beta } \right) \to \forall y\left( \alpha \right)} \right)$$
B
$$\forall x\left( {\forall z\left( \beta \right) \to \exists y\left( {\neg \alpha } \right)} \right)$$
C
$$\forall x\left( {\forall y\left( \alpha \right) \to \exists z\left( {\neg \beta } \right)} \right)$$
D
$$\forall x\left( {\exists y\left( {\neg \alpha } \right) \to \exists z\left( {\neg \beta } \right)} \right)$$
2
GATE CSE 2011
+2
-0.6
Which one of the following options is correct given three positive integers $$x, y$$ and $$z$$, and a predicate
$$P\left( x \right) = \neg \left( {x = 1} \right) \wedge \forall y\left( {\exists z\left( {x = y * z} \right) \Rightarrow \left( {y = x} \right) \vee \left( {y = 1} \right)} \right)$$
A
$$P(x)$$ being true means that $$x$$ is a prime number
B
$$P(x)$$ being true means that $$x$$ is a number other than 1
C
$$P(x)$$ is always true irrespective of the value of $$x$$
D
$$P(x)$$ being true means that $$x$$ has exactly two factors other than 1 and $$x$$
3
GATE CSE 2010
+2
-0.6
Suppose the predicate $$F(x,y,t)$$ is used to represent the statements that person $$x$$ can fool person $$y$$ at time $$t$$. Which one of the statements below expresses best the meaning of the formula $$\forall x\exists y\exists t\left( {\neg F\left( {x,y,t} \right)} \right)?$$
A
Everyone can fool some person at some time.
B
No one can fool everyone all the time.
C
Everyone cannot fool some person all the time.
D
No one can fool some person at some time.
4
GATE CSE 2009
+2
-0.6
Which one of the following is the most appropriate logical formula to represent the statement:

"$$Gold\,and\,silver\,ornaments\,are\,precious$$"

The following notations are used:
$$G\left( x \right):\,\,x$$ is a gold ornament.
$$S\left( x \right):\,\,x$$ is a silver ornament.
$$P\left( x \right):\,\,x$$ is precious.

A
$$\forall x\left( {P\left( x \right) \to \left( {G\left( x \right) \wedge S\left( x \right)} \right)} \right)$$
B
$$\forall x\left( {\left( {G\left( x \right) \wedge S\left( x \right)} \right) \to P\left( x \right)} \right)$$
C
$$\exists x\left( {\left( {G\left( x \right) \wedge S\left( x \right)} \right) \to P\left( x \right)} \right)$$
D
$$\forall x\left( {\left( {G\left( x \right) \vee S\left( x \right)} \right) \to P\left( x \right)} \right)$$
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