Taking the maximal number amongst the parameters.
$\max\{x_1,x_2\} = \cases{x_1, \text{if }x_1 > x_2\\x_2, \text{otherwise}}$
You can define like that the maximum of any finitely many elements.
When the parameters are an infinite set of values, then it is implied that one of them is maximal (namely that there is a greatest one, unlike the set $\{-\frac{1}{n} | n\in\mathbb{N}\}$ where there is no greatest element)
The symbol $\forall$ means "for all." The symbol $\in$ means "in" or is an element of.
In your context, it means for all $a$ and $b$ in $S$.
Best Answer
It is borrowed from computer programming: it means that the item on the left hand side is being defined to be what is on the right hand side. For example, $$y := 7x+2$$ means that $y$ is defined to be $7x+2$.
This is different from, say, writing $$1 = \sin^2(\theta) + \cos^2(\theta)$$ where we are saying that the two sides are equal, but we are not defining "1" to be the expression "$\sin^2(\theta) + \cos^2(\theta)$".
Basically, some people think that there should be notational difference between saying "I define
blah
to be equal toblankety
" and saying "blah
is equal toblankety
". So they use:=
for the first and=
for the latter. Usually, it is clear from context which of the two uses of the equal sign is intended (often because of signal words like "Let", "We define", etc.)