Is the function max{x,y} defined if x and y take equal values?

Question

If x and y take the same values, will the function return a result? I am asking this as maximum means greatest of two values. So if both the values are equal, the existance of the function confuses me.

Answer 1

The definition of the maximum of an ordered set $X$ is the following:

$m=\max(X)$ if $m\in X$ and $m$ is an upper bound for $X$.

Therefore, if $x=y$, then the set $\{x,y\}$ is a singleton and its maximum is (trivally) its only element.

Answer 2

$\max\{x,y\}$ is either $x$ or $y$. When $x=y,$ it doesn't matter which one you take; the result will be the same. So $\max\{x,x\}=x.$

Answer 3

The $\max$ function returns the maximal value of the two numbers $x,y$ (or more). If $x=y$, then so be it. Defining the function so that the arguments have to contain a unique maximal number seems unneccessary.