What do you get with sets that you don't get with plain logic? I am presently learning the basics of math, and I see a lot of proofs about things like unions and intersections. Every time something is proven about sets, the method is to translate statements like $S \cup R$ into $x \in S \lor x \in R$. As far as I can tell, $x \in S$ is nothing more than a proposition that might as well be written $S(x)$. Are sets just a notational shorthand for otherwise unstructured propositions about variables, or do they do anything special?
 A: Sets both organize information and delineate the fundamental elements of your system by defining the basic terms of the logical system. 
In a system of axioms and definitions, you can't define everything. You get an infinite regress if you try. Here's one of Euclid's attempts at defining a Point: "That which has no part." 
What's a point? A line? That second gets complicated in non-euclidean geometry. What's a great circle to Euclid is a line to Gauss. 
Formal logic has its propositions and universal qualifiers. Those don't define sets though or points or any entities to which to apply logic.  
Sets get your foot in the door for applying logic to various notions of grouping objects together. 
A: In any axiomatization of mathematics,  ZFC or NBG (for example) , the inference apparatus (the way you reason about the objects you work with) is logical in nature, but the objects themselves cannot be defined by pure logic. You need to define those objects in a different framework.  Not all operations involving the primary objects have simple  logical interpretations. Consider the axiom of choice, or the continuum hypothesis (and a couple  others).
A: Suppose you know that a number n is a rational number, sure you can express this as : 
Q(n)  , with  Q as predicate meaning " beind g a rational. 
Now, you want to infer from this that n is a real number, that is : 
R(n) with R as predicate meaning " being a real number". 
How could you justify your claim? 
Maybe your could reason intensionally ( conceptually) , as did aristotelian logic, and prove that the predicate " being rational" contains as  conceptual element ( conceptual "note" as said ancient logicians) the predicate " being real", as a species contains the concept of its genus. ( For example, the concept of " man" contains the concept of " animal", for " man" is a species of " animal"). 
But this conceptual approach might be difficult. 
Sets allow you an extensional approach. Instead of dealing with the concepts that express your predicates, you will deal with the denotations of this predicates, that is, with the set of objects that satisfy these predicates. So you will translate Q(n) for example as :
                 n belongs to { x | x satisfy the predicate Q} 

So , to prove that Q(n) implies R(n), you will show that the set of rationals is contained in the set of reals. 
Remark: note that the sense of the " is contained" relation is changed when we go from the intensional/conceptual approach to the extensional/set-theoretic approach. 
Your reasoning will be as follows: 
(1) The set of rationals is a subset of the set of reals, that is : 
for all x, if x belongs to Rationals, then x belongs to reals. 
(2) So, if n belongs to rationals, then n belongs to reals. 
(3) n is a rational. 
(4) Therefore, n belongs to the set of real numbers. 
And finally you will translate your conclusion in subject/predicate form : R(n). 
