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54 votes
54 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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Answer: B.

The given logical expression says that

“For Every person A, there is some person B, such that at some time T, A can Not fool B at T.”

For everyone we can find someone whom at some point of time they cannot fool.

So, No one can fool everyone at all times.

More easily, we can write the given logical expression in Equivalent form using De-Morgan’s law, as
$ ¬ ∃x∀y∀t(F(x, y,t)) $ which says that “Noone can fool everyone at all times.”

$\color{red}{\text{Analysis of Option D:}}$

Option D is saying that Noone can fool anyone at any time.

It is like saying that: in a group of friends, “Noone can betray anyone at any time.”

Logical Expression:
$¬ ∃x∃y∃t(F(x, y,t))$ ; which is Not equivalent to the given logical expression in the question.

$\color{red}{\text{Analysis of Option A:}}$ $\color{}{\text{}}$

Option A says that “Everyone can fool some person at some time”, for which logical expression will be:

$∀x∃y∃t(F(x, y,t))$ ; which is Not equivalent to the given logical expression in the question.

Statement in Option C seems ambiguous(interpretation ambiguity) when reading it.
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4 Answers

101 votes
101 votes
Best answer

$F(x, y, t) \implies$ 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.

edited by

4 Comments

No one means ~$\exists$
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Thanks
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Option c can be an answer if we change the statement(option c) from “all the time“ to “at some time”.

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46 votes
46 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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3 Comments

I'm not getting this: by your logic: ! ∃x∀y∀z(F(x,y,z) without negation: some person can fool everyone all the time.  
If you negate the sentence: some person can't fool everyone all the time.  so I'm not able to arrive at option b. pls help. 

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¬(∃x∀y∀t(F(x,y,t)))≡¬(∃x∀y∀t(F(x,y,t))). means that "it is false that there exists at least some x,which can fool all persons (y's) at all times". That means no 'x' exists which do it i.e "No one can fool everyone all the time"
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@Bhagirathi

Can you please check your $\mathbf{3^{rd}}$ step.

Where is the inner $\mathbf{Negation}$.

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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.

4 Comments

@arjun sir..... So the answer could be C also?
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Nopes. Then "all the time" in C should be changed to "some time".
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Thanks sir
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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)

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