Help understanding what is asked of me Is there a "min(a,b)" in math I do not know about? PLEASE DO NOT ANSWER THE QUESTION. I just need to know about "min" and what it stands for so I can figure out the question. 
Question:

Use a proof by cases to show that $\min(a,\min(b, c))=\min(\min(a, b), c)$ whenever $a$,$b$, and $c$ are real numbers.

 A: The function $\min(a,b)$ returns the minimum of $a$ and $b$: if $a<b$, then $\min(a,b)=a$; if $a>b$, then $\min(a,b)=b$; and if $a=b$, then $\min(a,b)=a=b$. You’re asked to prove that this function has a certain property.
You can, if you wish, think of it as a binary operation on real numbers; in fact, it’s sometimes written that way, using the symbol $\land$, so that $\min(a,b)=a\land b$.
A: In general for any subset of reals $A$ we can define $\inf A$ as a number $x\in\mathbb R$ such that
$$\forall a\in A, x\le a\quad\text{and}\quad\forall y((\forall a\in A, y\le a)\implies y\le x)$$
if it exists, it's unique. We define $\min S$ as $\inf S$ if $\inf S\in S$.
It's easy to see that for finite $S$, min always exists. Then your question becomes obvious (using the universal properties of the infimum above):
$$x=\min\{a,\min\{b,c\}\}\implies x\le a\land x\le\min\{b,c\}\implies x\le a\land x\le b\land x\le c\\\implies x\le\min\{a,b\}\land x\le z\implies x\le\min\{\min\{a,b\},c\}$$
Therefore $\min\{a,\min\{b,c\}\}\le\min\{\min\{a,b\},c\}$ and the other direction is analogous.
