I keep seeing $\max ( \cdot ) $ and $ \min ( \cdot ) $ everywhere.
EDIT :
It appears like this, $\max(4,9]$ or $\min[a,b)$ or sometimes with curly brackets, $\max \{0,100 \}$ etc.
I can sort of guess what it means, but I don't know what they are...
First of all, what is it even called? Is it a property of set, or a function?
Is it the same as supremum and infimum thing from Real Analysis?
PS : Idk the correct tag for this..
Thank you :)
$\endgroup$ 52 Answers
$\begingroup$If $S$ denotes a set and is equipped with a partial order $\leq$, and $A\subseteq S$ then $\max(A)$ exists if $A$ has an element $a$ such that $b\leq a$ is true for every $b\in A$.
In that case $a$ is unique and $\max(A):=a$.
In many cases such an element does not exist while a supremum does exist.
For instance $A=(0,1)\subseteq\mathbb R$ has $1$ as supremum, but it has no maximum. This because $1\notin A$.
Similar story for $\min(A)$.
$\endgroup$ $\begingroup$The maximum of a totally ordered set is defined as an element that is greater than all the other elements. For example $\max(4,9] = 9$ since $9$ is in $(4,9]$ and is greater than all the other elements. On the other hand, $\max(4,9)$ does not exist. You can keep picking bigger and bigger elements, closer and closer to $9,$ but none of these is the maximum, since you can always find something a little closer.
Minimum is the same idea, of course. Just replace 'greater' with 'less than'.
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