Difference between universal and existential statement for isosceles triangle.

Question

I know the obvious that for a universal statement usually it's: "For all [...]" and existential it's: "There exists [...]".

But this statement doesn't put it that obvious.

I have a feeling that is it a universal.

This is the statement:

Proposition. Let $XYZ$ be a right triangle with sides of length $x$ and $y$ and hypotenuse $z$. If the area is $z^2 /4$, then the triangle is isosceles.

Could someone shine some light on this please?

Answer 1

You're right. The theorem means exactly the same as:

Proposition. For every right triangle $XYZ$ with sides of length $x$ and $y$ and hypotenuse $z$, it holds that if the area is $z^2 /4$, then the triangle is isosceles.

The wording "Let $\langle\text{variable}\rangle$ be a $\langle\text{something}\rangle$" is just an alternative way to phrase "for all...", which is often used when stating theorems in order to make the prose flow better (because "Let ..." makes a complete sentence that you can follow with a full stop, whereas "For all ..." is a dependent clause that needs to be joined to the main statement by a comma).

Formally there's no difference between the two.

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Beware, however, of the difference to the phrasing

Let $k$ be $\frac{n+3}2 + p$.

which might appear inside a proof. The latter is neither a universal or an existential, but a definition: It simply gives a name to something that it is already fully clear what is. Note that in this case there's no "a" after "be", which signals that nobody has any choice for what to make $k$ means -- it has to be the precise number that is $\frac{n+3}2+p$, and nothing else.