Defining uniqueness with quantifiers

Question

I am trying to create a set of 10-tuples, where each tuple consists of elements taken from a set {defective, nondefective). Each sample must have precisely 1 defective part and 9 nondefective parts. My attempt

Let the sample be a 10-tuple $$S:=(X,\Omega,F)$$ $$X=\{i \ \epsilon \ Z:1\leq i\leq10\}$$

$$\Omega=\{\text{defective, nondefective}\} $$

$$F:X\to\Omega$$

$$Q=\{x \ \epsilon \ S:(\exists i \ \epsilon \ X) F(i)=\text{defective} \ \land \ (\forall n \ \epsilon \ X)n\ne i \to \ F(n)=\text{nondefective}\}$$

Assuming my notation is correct, the issue I see is that my quantifier for i does not preclude the existence of multiple defective parts being part of the tuple. What quantifier can I use that specifies existence and uniqueness?

Answer 1

Exists exactly one, notated by $\exists!$,
or perhaps $\exists$1, $\exists_1$ or whatever.
The one I'm familiar with is E!.

Exists exactly one x with P(x) means
exists x with (P(x) & for all y, (P(y) implies x = y)),
among other equivalent statements.

Instead of the uniqueness quantifier, one could use one of the equivalent statements.

Answer 2

While the symbol $\exists_1$ is commonly used, it is not part of the basic logic quantifiers. I fact, it is not needed, as uniqueness can be expressed as $$ \forall x,y : (P(x) \wedge P(y) \implies x = y).$$

Together, the existence of a unique $x$ with the property $P(x)$ may be written as $$(\exists x : P(x)) \wedge (\forall x,y : (P(x) \wedge P(y) \implies x = y)).$$