Is this definition of a function correct?

Question

So im trying to teach myself some logic and how quantifier scope and bound works and I saw the following defintion of a function which left me scratching my head.

$$\forall a\in A \exists b\in B:((a,b)\in R\land \forall b'\in B:(a,b')\in R\implies b=b').$$

We can see from the defintion that the variables $a$ and $b$ are bound in the entire expression by the brackets however my confusion arises for the variable $b'$ clearly it is bounded in the set $B$ however for the implication no brackets are used to bound $b'$ in the implication so basically my question is should we instead write

$$\forall a\in A \exists b\in B:((a,b)\in R\land \forall b'\in B:((a,b')\in R\implies b=b')).$$

With the addition of the extra brackets we have bounded the $b'$ in the implication as it falls within the quantifiers scope is this correct unnecessary or wrong?.

Thanks in advance.

Answer 1

Should we instead write

$$\forall a\in A \exists b\in B:((a,b)\in R\land \forall b'\in B:((a,b')\in R\implies b=b')).$$

Yes. You would use this to prove that the set of ordered pairs $R$ is a function mapping each element of $A$ to a unique element of $B$.

If you have established or simply assumed that $R$ is a function mapping each element of $A$ to a unique element of $B$, you could write

$$\forall a\in A: (R(a)\in B)$$

where $R(a)$ is the image of $a$ under $R$.

Answer 2

I'd say that your version is actually better. In the context of the definition of functions, where the author is establishing the uniqueness of $R(a)$ using Russell's definite descriptions, one should understand the last quantifier to range over the whole following conditional, but as written it could be interpreted as ranging over the antecedent only, and leaving the consequent with a free variable.