How to write "at least n" and "exactly n" in logical notation?

Question

$$ \exists x \exists y \exists z (x\neq y \land x \neq z \land y \neq z \land P(x) \land P(y) \land P(z)) $$

I would translate the above sentence as, "There exist 3 different elements (x,y,z) for which P(x), P(y), and P(z) are true."

Does this mean at least 3 elements or exactly 3 elements? I think it's at least 3.

If so, how do I formally write exactly 3 elements?

Answer 1

$P$ holds for at least three elements: $$\exists x\exists y\exists z\ [x\ne y\land x\ne z\land y\ne z\land P(x)\land P(y)\land P(z)]$$ $P$ holds for at most three elements: $$\forall w\forall x\forall y\forall z\ [P(w)\land P(x)\land P(y)\land P(z)\to (w=x\lor w=y\lor w=z\lor x=y\lor x=z\lor y=z)]$$ $P$ holds for exactly three elements: $$\exists x\exists y\exists z\forall w\ [x\ne y\land x\ne z\land y\ne z\land (P(w)\leftrightarrow(w=x\lor w=y\lor w=z))]$$

Answer 2

It says "at least three elements". Imagine a structure with four elements in which $P$ is true for all of them; clearly the sentence is satisfied in this structure.

A quick way to say "precisely three" is "at least three, and not at least four".

Answer 3

Something like $$\forall a\,\big(P(a)\Rightarrow (a=x)\vee (a=y)\vee (a=z)\big)$$ says that the property $P(a)$ is only satisfied if $a=x,y,z$.

Answer 4

First establish a set B to which $x,y,z$ may belong then

\begin{equation} \exists A\subset B\,(\,\vert A\vert=3 \land (\wedge_{x\in A}P(x)\,) \end{equation}

Answer 5

Okay, no-ones suggesting this and maybe it isn't valid but...

isn't $\exists ! a$ meaning there exists a unique $a$ acceptable, and....

isn't $\exists \{a,b,c\}$ meaning there exists a set f three elements acceptable?

If so (and maybe those notations aren't acceptable) wouldn't $\exists! \{x,y,z\} ...$ do it? There is a unique set of three elements where the conditions hold so there are three elements. If there were 4 or more elements where the conditions hold and we can make multiple different sets of 3 where the conditions hold. But as any set of three must be unique, there can only be those three elements.

Answer 6

$$ \exists x \exists y \exists z (x\neq y \land x \neq z \land y \neq z \land P(x) \land P(y) \land P(z)) \tag1$$

I would translate the above sentence as, "There exist 3 different elements (x,y,z) for which P(x), P(y), and P(z) are true." Does this mean at least 3 elements or exactly 3 elements?

Sentence $(1)$ means that $P(x)$ is true of 3 particular objects, which is equivalent to saying that it is true of at least 3 objects. (Strictly speaking, $P(x)$ being true of 3 objects is also equivalent.)

1. Being true of 3 particular objects does not preclude them being duplicate indistinguishable objects, so arguably neither implies 3 different objects nor implies 3 distinct objects.

   (On the other hand, 'different'/'distinct' can be understood as 'separate', in which case '3 different objects', '3 distinct objects', '3 particular objects' and 'at least 3 objects' are interchangeable.)

2. Sentence $(1)$ has not actually identified any three particular objects; $x,y,z$ are just dummy variables.

3. Being true of 3 particular objects does not preclude being true of 7 particular objects, so does not imply being true of exactly 3 objects, which can be formalised as $$∃a\:∃b\:∃c\:\Big(a\ne b\,∧\, a\ne c\,∧\,b\ne c\,∧\,∀s\,\big(P(s)\,↔\,s= a\,\lor\, s=b\,\lor\, s=c\big)\Big).\tag2$$

Sentence $(1)$ is logically equivalent to this more concise sentence: $$∀a\:∀b\:∃s\:\big(s\ne a\,∧\,s\ne b\,∧\,P(s)\big).$$