First Order Logic

Question

Comments

i think all are correct?

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Ya D is the option. Can you explain why S1 is correct?

S2 is correct and S3 is the negation of S1.

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In above statement if everyone is not a politician then B converts to $false\; \rightarrow\; \exists_x Q(x)$ which is true but above statement yields false.

How is it right??

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Actual logic is $\forall xp(x)->\exists x Q(x)$

apply p->q=~pvq.

$\sim \forall (x)p(x) V\exists (x) Q(x)$

$\exists x \sim p(x) V\exists (x) Q(x)$

$\exists x[ \sim p(x) V Q(x)]$

$\exists x[ p(x) -> Q(x)]$

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Got it man!! Thanks a lot.

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@santhoshdevulapally

In above statement if everyone is not a politician then B converts to false→∃xQ(x) which is true but above statement yields false.

How is it right??

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instead of taking statement derive using predicates.

otherwise take p(1)= F and p(2)=T. & Q(1)=T and Q(2)=F.

$\forall xP(x)=p(1)\wedge p(2)

   \exists xP(x)=p(1)\vee p(2)$.

apply for p->q and q->p then u get p=q.

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https://gateoverflow.in/103060/logic