Use quantifiers to express:->There is exactly one person whom everybody loves

Question

Let L(x, y) be the statement "x loves y," where the do­main for both x and y consists of all people in the world. 
Use quantifiers to express: 
Q->There is exactly one person whom everybody loves.
E-some || A-for all ||
Answer-> Ex ( AyL(y,x) ^ Az((Aw L(w,z))->z=x) )
Can anyone explain the answer and how to solve these questions
i mean how to analyze these questions??

Answer 1

First of all we have only two quantifiers- exists and forall. So, first try to convert the given sentence into a sentence containing those (its just English)

There is exactly one person whom everybody loves

This can be rewritten as

There is someone who is loved by everyone, and there is no one else who is loved by everyone

Again translated as:

There is someone, x, who is loved by everyone, and  if there is another one, y,  who is loved by everyone else, means x and y are the same person.

This can be translated into first order logic (first order logic is just propositional logic with ∃ and ∀)

∃x(∀yL(y,x) ⋀  (∀z(∀wL(w,z)) ⇨ (z = x)))

Here actually in place of w, we can use y as the life of y expires immediately after ∀yL(y,x) this. Using w instead of y is strictly not needed. 

You can also try to express "There is at most one person whom everybody loves" 

Now, the translations I did is based on my knowledge about the expressive powers of first order logic. That is first order logic can't be used for all statement forms- we first need to convert it into a suitable form. Luckily there are only a few cases, which you should get experienced by solving previous year questions in this topic. You can see the below 3 questions and they should cover almost any type of question from this topic. 

https://gateoverflow.in/256/gate1992_92-xv

https://gateoverflow.in/923/gate2003_33

https://gateoverflow.in/922/gate2003_32

Comments

I have read about the 3rd quantifier(kenneth rosen) which correcponds to "there is an unique x for which..." i think we can use it for this type of problem

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Sir i think ∀z is written instead of ∃z in ∃x(∀y(y,x) ⋀  (∃z(∀w(w,x)) ⇨ (z = x))),for here it meant For any. This is used in many solutions given in keneth h rosen book.

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Yes, it should be $\forall$ only otherwise the meaning goes wrong. I have corrected.

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∃x(∀y(y,x) ⋀  (∀z(∀w(w,x)) ⇨ (z = x)))

Here should it not be like ->

∃x(∀y(y,x) ⋀  (∃z(∀w(w,z)) ⇨ (z = x))) ?

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no it should  be for every z instead of there exist z

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@Arjun sir

Why can't it be $\exists x(\forall y(y,x)\wedge \forall z(L(y,z)\rightarrow (z=x)))$ ?

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It would mean, Every y loves x and only x.

Which will in turn mean that no one else is loved by anyone but the question does not prohibit it.

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but in the questions its mentioned everybody loves exactly one person which is x

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Can you make a set of 3 people and show the relationship such that the given logic is satisfied?

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Arjun sir

I'm having difficulty in understanding the question

Everybody loves exactly one person

Does this means there is exactly one person x whom everybody loves? Eg-among 4 persons A, B, C, D. all loves D

or everybody loves one person. Eg-if there are 4 persons A, B, C, D

A loves B, B loves A, C loves D and D loves C?

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Yes, Thats the only confusion you have. Being an English sentence you can clear this with some good English person.

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:(

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I meant that is the ambiguity of English language and you are correct only. The two interpretations are possible here. As long as you can do examples like above you'll be able to answer correctly in GATE.

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Is this also correct?

∃x(∀yL(y,x) ⋀  ∀z( (x $\neq$ z) $\rightarrow$  ∃w~L(w,z) ) )  i.e there exists some x whom everybody loves and for all z such that if x and z are not same then there exists atleast one person w who doesnt love z.