In set theory, we distinguish between "naive" set theoy vs. "axiomatic" set theory. Naive set theory involves the exposition of the relevant concepts of set theory in informal language. Paul R. Halmos, in his famous work "Naive Set Theory", describes his book in its preface as "axiomatic set theory from the naive point of view. It is axiomatic in that some axioms for set tehory are stated and used as the basis for all subsequent proofs. It is naive in that the language and notation are those of ordinary informal (but formalizable mathematics"(Halmos). Thus, while a great many important axioms are explained in his work, the means of exposition is informal language rather than purely dense (and condensed) higher-order logic notation. The work thus becomes accessible for those whose knowledge of mathematics is minimal.
What is helpful about 'naive' approaches is that they are borderline-conversational in their informality. Of course, such approaches are limited, and the serious student of set theory must at some point acquire knowledge of the semantics and syntax of higher-order formal logic. But why is set theory so useful? Because, as Halmos states, "The mathematical concept of a set [the most central concept of set theory; hence the name] can be used as the foundation for all known mathematics"(Halmos). This makes set theory quite powerful. That said, let us delve into an outline of the basics of set theory.
A "set" is a collection of objects or "elements" which make up the "members" of a set. While "set" is oftentimes used interchangeably with that of "class", the beginning student ought to get accustomed to using "set" for now, as "class" can sometimes be used with a distinct, technical name. In more advanced set theory, a "set" is not quite equivalent to a class. One can conceive of a set of just about anything. For example, a "set" of all fruits might contain apples, pears, grapes, bananas and so on. We say that elements or members of a set "belong" to that set. If x is an apple, and an apple is a fruit, and A is the set of all fruits, then we say that x is a member of A. We would symbolize this by writing x ∈ A.
The symbol for belonging that we see here comes from the Greek letter "epsilon", from which we get the Latin letter "e." Whenever you see the symbol used, know that "belonging" is what is being symbolized. Keep in mind also that it is not necessarily the case that capital letters have a distinct meaning relative to lower-case letters. However, for our purposes, capital letters will be used to signify classes and lower-case letters will be used to signify their elements.
A symbol with which readers will likely be more readily familiar is the sign for equality. A = B means that set A is equal to set B. Indeed, this is not only a truth, but a central axiom of set theory. It is what is known as the Axiom of Extension. To give an informal definition of the Axiom of Extension: Two sets are equal to one another if and only if they have the same elements. This is also known as the Axiom of Extensionality.
For those who have studied higher-order logic, here is a formal definition of the Axiom of Extension or Axiom of Extensionality:
Axiom of Extensionality: ∀x∀y[∀z(z∈x ≡ z∈y) → x=y] (Jech, 2011)
To give a brief exposition of this concept: for every x and for every y, if z is a member of x if and only if z is a member of y, then set x is equal to set y. In the words of Halmos: "With greater pretentiousness and less clarity: a set is a determined by its extension." Likewise, we have the symbol for inequality in set theory. A ≠ B means that "set A is not equal to set B."
Next, we will look at inclusion symbolism. Let us take sets A and B. Next, suppose that every element of A is also an element of B (although this does not necessarily mean that the two sets have to be identical). Where B includes the elements of A, we say that A is a subset of B. We would symbolize this with "A ⊂ B." We could also phrase this by saying that B includes A, in which case we would write "B ⊃ A."
Counterintuitive though it may sound, we must keep in mind that each set is included itself. Therefore, A ⊂ A. The fact that teach set is included in itself is known as the property of reflexivity. Therefore, sets possess the property of reflexivity because they are included in themselves. A is a subset of A.
It is also the case that equality necessarily possesses the property of reflexivity. For two sets to be equal is for each to be included as a subset of the other. Whenever we say that a set is a subset of another without being equal to it, we say that the inclusion or the subset is "proper." Therefore, if A ⊂ B but A ≠ B, the inclusion is "proper." A is a proper subset of B.
Let's look at the property of transitivity. Where A ⊂ B and B ⊂ C, then A ⊂ C, then the set inclusion being observed possesses the property of transitivity. Equality likewise possesses the property of transitivity, just as it does reflexivity.
Next, we will look at conditions of set inclusion which exhibit the property of equality, which, as we have seen, is made possible by the Axiom of Extension. Where A ⊂ B and B ⊂ A, A = B. We describe this by saying that the set inclusion being observed is antisymmetric. This is quite different from equality. Equality possesses symmetry. It is "symmetric" because if A = B then B = A. In the words of Halmos, "The axiom of extension can...be reformulated in these terms: IF A and B are sets, then a necessary and sufficient condition that A = B is that both A ⊂ B and B ⊂ A. Correspondingly, almost all proofs of equalities between two sets A and B are split into two parts; first show that A ⊂ B, and then show that B ⊂ A"(Halmos).
Halmos points out the differences between the properties of inclusion and that of membership. They sound similar, but there are important differences. He notes that inclusion is always reflexive. But we cannot say this of membership. Likewise, it is always the case that A is a subset of A. But this does not mean that we can say that the set A is an element of A. A is the set, not an element of itself. Finally, although inclusion is transitive, this is not true of membership or belonging.
Halmos, Paul. "Naive Set Theory." Litton Educational Publishing, 1960.
Jech, Thomas, "Set Theory", The Stanford Encyclopedia of Philosophy (Winter 2011 Edition), Edward N. Zalta (ed.), URL = <plato.stanford.edu/archives/win2011/entries/set-theory/>.