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) ? $0$ ?
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Maximum need to be a particular element of the set.
In this case, it doesn't exists.
Hence maximum doesn't exists.
Siong Thye Goh
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what if we assoiate max to a function such that $$g(L) = max{x_i:(x_i,f_i)∈S,f_i=L}$$ – younes zeboudj Jul 03 '17 at 22:46
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1Your function is not well defined then. you might want to handle the cases of empty set separately. – Siong Thye Goh Jul 03 '17 at 22:54
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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.
ZTaylor
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