Predicate Logic Translation, Implications and singularity

Question

So I've tried to translate two sentences, each with the predicates:

$B(x,y): x\text{ beats }y$

$F(x): x\text{ is a football team}$

$Q(x,y): x\text{ is a quarterback for } y$

$L(x,y): x\text{ loses to }y$

$c: Wildcats$

$j: Jayhawks$

I suddenly became a bit nervous about the translations, and had some questions about my translation.

1. Every football team has a quarterback $$\forall x(F(x)\to \exists y(Q(y,x))) $$ My predicate logic formula would translate to: "For all football teams, there exists a quarterback". Should I use $\to$ or should I translate it with an $\land$ instead. Another question is couldn't this be understood as if all the football teams had the same quarterback?
2. The Wildcats beat some team, which beat the Jayhawks $$\exists x(F(x)\land B(c,x)\land B(x,j))$$ Is this an okay translation or should I use implies?

After reading the book it seems like $\to$ is used frequently if not always with the universal quantifier $\forall$, while $\land$ is used when we have the existential quantifier $\exists$. Is this pattern true? I've never seen anyone mention it, and there is nothing of it in the book. Is there a good way to phrase the translation, that it might be clearer if it should be $\land$ or $\to$?

The question about the singularity or multiplicity of the existential quantifier. How should I differentiate between:

• It is one and the same quarterback for all football teams
• All football teams have one quarterback,
• All football have some who are their quarterback (a group of quarterbacks who all are quarterbacking for all teams)
• All football have some who are their quarterback (each football team has their own group of quarterbacks)

When searching for the answers it seems like there is the quantifier $\exists !$ which should only mean one, but that is not part of the language I've been taught.

Answer 1

With respect to the last part of your question, i have tried and answered them here

You can use the $\exists!$ quantifier, which reads as "there exists a unique". You can read more here, which shows how to represent the symbol in the symbols you are "allowed" to use.

• If you want to say that all the teams have the same quarterback, you can put a $\exists! y$ in 1. out front, i.e. write $\forall x \exists y!(F(x) \to Q(y,x))$
• Use the $\exists!$ quantifier. I.e. $\forall x (F(x) \to \exists! y Q(y,x))$
• I'm not too sure what you are saying, but maybe this works $\forall x \forall y(F(x) \land (\exists z)Q(y,z) \to Q(y,x))$. i.e. all quarterbacks are quarterbacking all teams
• I am not too sure, but $(\forall y) (\exists! x) Q(y,x)$ means that all quarterbacks only quarterback one team (but it also means that there are no quarterbacks that do not have a team, which i am not sure you want?).