TITLE: Upper and Lower Bounds of $\emptyset$
QUESTION [17 upvotes]: From some reading, I've noticed that $\sup(\emptyset)=\min(S)$, but $\inf(\emptyset)=\max(S)$, given that $\min(S)$ and $\max(S)$ exist, where $S$ is the universe in which one is working. Is there some inherent reasoning/proof as to why this is? It seems strange to me that an upper bound of a set would be smaller than a lower bound of the same set.

REPLY [3 votes]: A very natural explanation of the correct definition for the values of min and max on empty sets arises from their dual universal definitions - analogous to the universal GCD, LCM definitions that I presented in a post here.
First some notation. $\;$ Write $\ \ \rm x \le S \;\iff\; x \le s,\;\: \forall\: s \in S,\;$ and dually for $\rm\; x \ge S$
DEFINITION of $\:$ min $\quad$ $\quad\rm x \le S \;\iff\; x \le min\ S$
DEFINITION of max $\:\quad$ $\quad\rm x \ge S \;\iff\; x \ge max\ S$
For min, when $\;\rm S = \emptyset\;$ is empty, the first clause  $\;\rm x \le S\;$ in the min definition is vacuously true, hence the definition reduces to $\;\rm x \le \min \emptyset\;$ for all $\;\rm x\;$. Hence $\;\rm \min \emptyset = \infty\;$. Dually $\;\rm \max\emptyset = -\infty\;$. 
As I remarked in said GCD, LCM post, such universal definitions often facilitate slick proofs. For some nontrivial examples of min, max flavor consider the slick proofs of the integrality of various products of binomial coefficients by employing the floor function, e.g. see Joe Roberts: Elementary Number Theory. Instead I close with an analogous slick GCD, LCM proof from my mentioned post (see it for further details).
Generally, in any domain, we have the following dual universal definitions of LCM and GCD:
DEFINITION of LCM $\quad$ If $\quad\rm a,b\ |\ c \;\iff\; [a,b]\ |\ c \quad$ then $\quad\rm [a,b] \;\;$ is an LCM of $\;\rm a,b$
DEFINITION of GCD $\quad$ If $\quad\rm c\ |\ a,b \;\iff\; c\ |\ (a,b) \quad$ then $\quad\rm (a,b) \;$ is an GCD of $\;\;\rm a,b$
Note: that $\;\rm a,b\ |\ [a,b] \;$ follows by putting $\;\rm c = [a,b] \;$ in the definition. Dually $\;\rm (a,b)\ |\ a,b \;$
Such $\iff$ definitions provide slick unified proofs of both arrow directions, e.g. the fundamental
THEOREM  $\rm\quad (a,b)\ =\ ab/[a,b] \;\;$  if $\;\rm\ [a,b] \;$  exists.
Proof:  $\rm\quad\quad d\ |\ a,b \;\iff\; a,b\ |\ ab/d \;\iff\; [a,b]\ |\ ab/d \;\iff\; d\ |\ ab/[a,b] \quad\;\;$ QED
The conciseness of this proof arises by exploiting to the hilt the $\iff$ definition of LCM and GCD.  Compare to less concise / general / illuminating proofs in many number theory textbooks.