What does $\max$ function returns as value when the set provided is empty?

Question

Having :

$$S= \{ (2,a), (2,l), (1, h), (7,s), (7,a) \}$$

what does this return as value :

$$\max\{x_i : (x_i, f_i) \in S , f_i=c \}$$

in other words, what does $\max$ returns when the condition is not satisfied (no element available to test) ? $0$ ?

Answer 1

Maximum need to be a particular element of the set.

In this case, it doesn't exists.

Hence maximum doesn't exists.

Answer 2

Convention is to write the supremum of the empty set as $-\infty$ and the infimum as $\infty$. (Note that $\sup X < \inf X$ if and only if $X = \emptyset$.) The maximum is the supremum if the set in question contains the supremum, and does not exist otherwise. Thus, the maximum of the empty set does not exist.