Homework: Existential Unique Quantifier

Has anyone looked at the homework yet? First of all, a whole new quantifier is introduced: ∃!

For an open sentence p(x) and a set S, “(∃! x ∈ S)(p(x))” means:

(∃ x ∈ S)(p(x)) and (∀ u,v ∈ S)([p(u) and p(v)] ⇒u = v)

In very basic terms, this is the “there exists one and only one” quantifier. The definition states that for any element, x, in a set that satisfies an open statement, any other element that does the same must be equal to that element, x.

The last question for the homework was pretty interesting as well.

Quantify the statement: There is no largest integer. 

I decided to quantify this by saying something like, there will always be an integer bigger than the one you are looking at in any case. Hopefully that means the same thing? I ended up with:

(∃ x ∈ ℤ) (∀ y ∈ ℤ) (x > y)

Happy Friday, and good luck with the homework!


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