made easy test series

Question

Consider the following predicates:

S(x): x is a student

GATE(x,y): x has written gate in stream y.

Which of the following is equivalent predicate logic for the statement : “There doesnt exist a student who has written GATE in every stream.”

(a)$\exists y\exists x[S(x)\Lambda \sim GATE(x,y)]$

(b)$\forall y\exists x[\sim S(x)V \sim GATE(x,y)]$

(c)$\forall x\exists y[\sim S(x)V \sim GATE(x,y)]$

(d)$\exists y\exists x[\sim S(x)\Lambda \sim GATE(x,y)]$

Comments

No. answer has been given as option (C). But I think option (a) is correct..

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Yes $C$ should be correct, my bad

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But why?Can u pls explain? $\sim S(x)$ implies "not a student"..but he/she needs to be a student right? what is wrong with option a??

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answer should be

$\forall x\exists y\left [ S(x) \wedge \sim Gate(x,y))) \right ]$

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Option $a$ says some $y$ and some $x$ such that $x$ is a student and he does not writes in stream $y$.

$Ex:$ $(a,b,c)$ are students, $(u,v,w)$ are the streams, $a,b$ writes in all stream and $c$ writes in stream $(u,v)$

Now option $a$ will still be true as there exist some $y$ i.e w and there is some $x$ i.e. c such that student $x$ does not writes in $y$. But the statement "There doesnt exist a student who has written GATE in every stream." is not true as $a,b$ has written in all the streams.

 

$c)\,\,$$\forall x∃y(\neg S(x)\,V\,\neg Gate(x,y))$ $\equiv$ $\forall x∃y(S(x)\rightarrow\neg Gate(x,y))$

Now option $c$ says "For all $x$ there is some $y$ such that if $x$ is a student he has not written gate exam in stream $y$". So, this is equivalent to the required statement .

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@Shobhit Joshi

brother --> will not be there we need $\wedge$ there

according to your statement even if all x are not student :) still it will be true.

 

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Shobhit Joshi

Thanks..got it now :)

jatin khachane 1

If x isnt a student, then it doesnt matter whether it is true or false since the statement only speaks of students and not of non-students..so it may be true or may not be.

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@jatin khachane 1 if there is no student, then "There doesn't exist a student who has written GATE in every stream", this is also satisfied. Isn't it ?

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yes @Shobhit Joshi

 you right if not student then it may be true / false