# There Exists Only One

1. Jul 2, 2013

### Bashyboy

1. The problem statement, all variables and given/known data
Express each of these system specifications using predicates, quantifiers, and logical connectives, if necessary.

2. Relevant equations

3. 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?

2. Jul 2, 2013

### D H

Staff Emeritus
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, $\forall n(n\ne m \rightarrow \neg A(u,n))$.

3. Jul 2, 2013

### Bashyboy

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

4. Jul 2, 2013

### verty

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

5. Jul 8, 2013

### Bashyboy

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

6. Jul 8, 2013

### haruspex

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 $¬A(u, m_1)))$.