How to express that a set has only one maximum

Question

In a given set of numbers N there is one, only one, that is larger than all the others. What is the best way to express this?

My guess:

$\exists x \in N |( x > \forall y \in N \wedge \forall y \in N = x \Rightarrow y = x) $

or:

$\exists ! x \in N | x > \forall y \in N$

Is that correct or do I need to indicate that x is larger than all the elements except itself? For instance:

$\exists ! x \in N | x > \forall y \in N | y \neq x $

Answer 1

Writing $x > \forall y \in N$ is highly non-standard. It might be understood if you clarify elsewhere, or if it's clear from context, but you should probably not expect it. Similarly, using your $\mid$ as a sort of reverse implication outside of set notation is also non-standard.

The standard way to write this would be $$ \exists x \in N(\forall y(x \neq y \to x > y)). $$ Note that $x > y$ already precludes the possibility that there might be another $x'$ which is also greater than all other elements, so you don't need to express uniqueness separately.

Answer 2

How about

$$ \exists ! x \in N, \forall y \in N, x \neq y \implies x > y$$

Answer 3

Another possible option is "there is a unique element $x$ of $X$ such that $x = \max (X)$" $$\exists_{1} x \in X: \quad x = \max (X)$$

Note that one usually writes $\forall y: x \geq y$. Also note that the statement $$\exists x \in X \quad \forall y \in X: x > y$$ means that $x>x$, a constradiction to the axioms of equality relation. Therefore $$\exists x \in X \quad \forall y \in X/\{x\}: \quad x > y$$ or $$ \exists x \in X\quad \forall y \in X: \quad x \neq y \implies x > y$$ In two last cases you do not need to state the uniqueness of such $x$ anymore.