Express the statement in logical expression

Question

Express the statement "Everyone has exactly one best friend" as a logical expression involving predicates,quantifiers with a domain consisting of all people,and logical connectives without using uniqueness quantifier.

I am confused pleased explain it

Answer 1

Everyone has exactly one best friend.

Let me define few predicates before proceeding to the answer :

$F(x,y)$ = $y$ is a Best friend of $x$. And let the domain be All People in the world.

(Assuming that No one is a best friend of himself/herself. (i.e. $F(x,x)$ is False) )

$\forall x (\exists y F(x,y) \wedge \forall z ((z \neq y) \rightarrow \sim F(x,z)))$

Interpretation : For every person $x$ there is some person $y$ who is best friend of $x$ And for any(every) person $z$, if $z$ is not same person as $y$ then $z$ is Not a best friend of $x$.

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Refer here for clarity about Uniqueness Quantifier/Quantification : https://gateoverflow.in/219473/kenneth-rosen-ch-1-ex-1-5-qn-52?show=219480#a219480

Answer 2

Instead of including a new variable z we can also express the statement as $\forall x\exists !yB(x,y)$  where $\exists! $ is the "uniqueness quantifier".

Answer 3

people =p(x)

best friend=b(y)

∀x(p(x)⟶∃y(b(y)))

Comments

If $b(x,y)$ means $y$ is a best friend of $x$ and If the domain is taken to be All the people in the world then the expression ∀x(p(x)⟶∃y(b(x,y)))

For every person $x$, there is at least one person who is best friend of $x$.

It does not say "One and Only One".