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70 votes
70 votes

Suppose the predicate $F(x, y, t)$ is used to represent the statement that person $x$ can fool person $y$ at time $t$.

Which one of the statements below expresses best the meaning of the formula,

  $\qquad∀x∃y∃t(¬F(x,y,t))$ 

  1. Everyone can fool some person at some time
  2. No one can fool everyone all the time
  3. Everyone cannot fool some person all the time
  4. No one can fool some person at some time
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5 Answers

Best answer
123 votes
123 votes

$F(x, y, t) \implies$ person $x$ can fool person $y$ at time $t.$ 

For the sake of simplicity propagate negation sign outward by applying De Morgan's law.

$∀x∃y∃t(¬F(x,y,t)) \equiv ¬∃x∀y∀t(F(x, y, t))$ [By applying De Morgan's law.]

Now converting $¬∃x∀y∀t(F(x, y, t))$ to English is simple.

$\color{blue}{¬∃}x\color{green}{∀}y\color{red}{∀}t(F(x, y, t))\implies \color{blue}{\text{ There does not exist }}$$ \text{a person who can fool }$$\color{green}{\text{everyone}} $$\color{red}{\text{ all}} \text{ the time.}$

Which means No one can fool everyone all the time.

So, option (B) is correct.

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50 votes
50 votes
B is the correct answer. The trick is to bring the negate sign to the extreme left. Form a sentence without using negate and just negate that.

$\forall x \exists y \exists t(\neg F(x,y,t))$
$\equiv \neg(\neg\forall x \neg \exists y \neg \exists t) (\neg F(x,y,t))$
$\equiv \neg (\neg\forall x \neg \exists y \neg \exists t ( F(x,y,t)))$
$\equiv \neg (\exists x \forall y \forall t ( F(x,y,t)))$.
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11 votes
11 votes

Without negation the statement is like: Everyone can fool Someone at some time. 

So symply, with negation it will be like: No one can fool everyone all the time.

3 votes
3 votes

∀x∃y∃t(¬F(x,y,t))∀x∃y∃t(¬F(x,y,t)) 

Everyone cannot fool  someone at some time

   (∀x)             (¬F)           (∃y)              (∃t)

to translate in hindi in other words

har koyi kisi na kisi  ko bewakoof banane mein  asafal rahy ga kisi na kisi time pe

so

no one can fool everyone all the time

Answer(B)

Answer:

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