Basic Game Theory let $x$ be a generic act in a given set $F$ of feasible acts and let $f(x)$ be an index associated to (or appraising) $x$; then find those $x^{0}$ in F which yield the maximum (or minimum) index, i.e., $f(x^{0})$ greater than or equal to $f(x)$ for all $x$ in $F$.
I am not understanding what it means for $f$ of $x$ to be an 'index to or appraising $x$'
I am also confused about the notation used for $x^{0}$
   -is it the case the $x^{0}$ is just an indication of a single act?
   -is $x^{0}$ a part of a the set $x^{0}$,$x^{1}$,$x^{2}$...$x^{n}$?
I am also confused about how to interpret $f(x^{0})$ greater than or equal to $f(x)$
   -will the clarification for my second question help me interpret $f(x^{0})$
   -what is the difference between $f(x^{0})$ and $f(x)$
I apologize for my lack of knowledge... I am having a lot of problems interpreting the different symbols that are presented in books on probability, statistics and game theory.  I keep running into notation involving infinite sets and complex functions... is there a book that can provide an elementary introduction to infinite sets and/or complex functions? 
I have not began to take a calculus class and I have only began a pre-calculus class for my first semester in college.  I was hoping to get ahead by studying topics such as, probability statistics and game theory.
 A: I'm pretty sure $x^0$ is just meant to be an arbitrary variable denoting a member of the set $F$.  (Using a superscript like that feels strange to me, though; I'm more used to seeing $x_0$ used that way.)  All the superscript 0 is supposed to do is indicate that $x^0$ is not the same variable as plain $x$. The author could just as well have called it $y$ or $x'$ or whatever.
When I see notation like that used, there's usually a connotation that the super/subscript 0 indicates a constant: we are to find one fixed $x^0$ which satisfies the criterion $f(x^0) \ge f(x)$ for all $x$ in $F$.  Here, plain $x$ is a bound variable which has no definite value outside the scope of that statement, while $x^0$ is an unbound constant which is hereby defined and available for use later.
Essentially, we're "picking out one of the $x$'s" as special and assigning it the label "0".  (Note, though, that in the exercise you quoted, $x^0$ may not actually be uniquely defined.)
Of course, if there was need to single out another element of $F$, an author might well call if $x^1$ (or $x_1$, in the notation I'm more used to) and there could conceivably also be an $x^2$ (or $x_2$) and so on.  But merely defining $x^0$ doesn't really imply that any other sub/superscripted $x$'s would have to be defined too; the 0 is just a label, not part of a sequence.
As for "let $f(x)$ be an index associated to (or appraising) $x$", that's presumably just a funny way of saying that $f$ maps (or "associates") each $x$ to a number, which is presumably supposed to somehow "appraise" the action $x$ (such that the best action maps to the highest number).  "Index" here is basically just another word for "number".
A: "An index associated to (or appraising)" is not a mathematical term.  It's just saying $f$ is some function of $x$ that we want to maximize (or minimize).  $x^0$ is just a particular value of $x$ that you want to find.  The condition says that the value of $f$ at $x^0$ is the greatest possible value, i.e. it is greater than or equal to the values of $f$ at all possible $x$. 
