System Specifications: Every User Has Access to Exactly One Mailbox

[Bashyboy]

Homework Statement

Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary.

a) Every user has access to exactly one mailbox.

Homework Equations

The Attempt at a Solution

It is typical of my book to not answer questions as given with the unique existential quantifier $\exists !$. For instance, the answer to the question above is $∀u∃m(A(u, m)∧∀n(n \ne m→¬A(u, n)))$. However, I am not convinced that this form assures that only one m exists for every u. Isn't it still possible that $m_0$ and$m_1$ are two elements in the domain of the variable that make the statement, implying that there doesn't exists one and only one value of m for every u?

 

[D H]

For instance, the answer to the question above is $∀u∃m(A(u, m)∧∀n(n \ne m→¬A(u, n)))$. However, I am not convinced that this form assures that only one m exists for every u. Isn't it still possible that $m_0$ and$m_1$ are two elements in the domain of the variable that make the statement, implying that there doesn't exists one and only one value of m for every u?

No. If those m₀ and m₁ are distinct (i.e., m₀ ≠ m₁), then both of them cannot satisfy A(u,m₁) per the second part of the condition, $\forall n(n\ne m \rightarrow \neg A(u,n))$.

 

[Bashyboy]

Well, why couldn't every n correspond to m₀, and then every n also correspond to m₁?

 

[verty]

Well, why couldn't every n correspond to m₀, and then every n also correspond to m₁?

It ranges over EVERYTHING, everything in the universe of discourse (or at least, everything that it can represent).

 

[Bashyboy]

Verty, I am not certain how that aids in answering my question.

 

[haruspex]

Homework Statement

Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary.

a) Every user has access to exactly one mailbox.

Homework Equations

The Attempt at a Solution

It is typical of my book to not answer questions as given with the unique existential quantifier $\exists !$. For instance, the answer to the question above is $∀u∃m(A(u, m)∧∀n(n \ne m→¬A(u, n)))$. However, I am not convinced that this form assures that only one m exists for every u. Isn't it still possible that $m_0$ and$m_1$ are two elements in the domain of the variable that make the statement, implying that there doesn't exists one and only one value of m for every u?

You are guaranteed the existence of m₀, say, such that $A(u, m_0)∧∀n(n \ne m_0→¬A(u, n)))$. Suppose m₁ (≠m₀) satisfies $A(u, m_1)$. But we know $∀n(n \ne m_0→¬A(u, n)))$. Since n can be m₁, and $A(u, m_1)$, it follows that $¬A(u, m_1)))$.