Eli5: Unique existential quantification (Ǝ!)

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So I’m trying to figure out how unique existential quantification is read and where it is used in the world.

Part 1 of my eli5: I don’t understand the statement

ƎxP(x) is true when P(x) is true for at least one value of x.

Part 2 of my eli5: how does the unique existential quantification come in play?

Ǝ!

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In: 2

Anonymous 0 Comments

> I don’t understand the statement “ƎxP(x) is true when P(x) is true for at least one value of x.”

1. `Ǝ` is read, “There exists…”
2. `Ǝx` is read, “There exists `x` such that…”
3. `Ǝx P(x)` is read, “There exists `x` such that `P(x)`.”

A domain for `x` typically needs to be defined, or is implicit. To be more explicit, you might write `Ǝx∈X P(x)`, which means, “There exists an `x` in `X` such that `P(X)`.”

So for example, if `X` is the set of integers, and `P(x)` is the boolean-valued function that tells you if a number is prime or not, then the statement `Ǝx∈X P(x)` is true, because there does exist a an `x` in the set of integers such that `P(x)` (“`x` is prime”) is true.

> how does the unique existential quantification come in play?

Unique existential quantification just means that you’re stating there exists one and only one element that satisfies the condition.

So for example, `Ǝ!x∈R x>0` (where R is the set of real numbers) is false, because it is not true that there exists one and only one real number that is greater than zero.

Unique existential quantification is just some syntactic sugar for giving us a compact way to describe the unique existence of something.

`Ǝ!x∈X P(x)` could be rewritten just in terms of existential quantification and universal quantification. E.g., `Ǝx∈X (P(x) ∧ ∀y∈X y≠x → ¬P(y))`. I.e., “There exists an `x` in `X` such that `P(x)` **and** for all other members `y` of `X` that are not `x`, `P(y)` is not true.”