A set is a collection of things called elements of the set. For example, the set of integers, the collection of signed whole numbers such as 1,2,−4, etc. This set whose existence will be assumed is denoted by ℤ. Other sets could be the set of people in a family or the set of donuts in a display case at the store. Sometimes parentheses,
Axiom 1.1.1 Two sets are equal if and only if they have the same elements.
Axiom 1.1.2 To every set, A, and to every condition S
Axiom 1.1.3 For every collection of sets there exists a set that contains all the elements that belong to at least one set of the given collection. (You can take the union of a bunch of sets.)
Axiom 1.1.4 The Cartesian product of a nonempty family of nonempty sets is nonempty.
Axiom 1.1.5 If A is a set there exists a set, P
These axioms are referred to as the axiom of extension, axiom of specification, axiom of unions, axiom of choice, and axiom of powers respectively.
It seems fairly clear you should want to believe in the axiom of extension. It is merely saying, for example, that

In this notation, the colon is read as “such that” and in this case the condition is being a multiple of 2.
Another example of political interest, could be the set of all judges who are not judicial activists. I think you can see this last is not a very precise condition since there is no way to determine to everyone’s satisfaction whether a given judge is an activist. Also, just because something is grammatically correct does not mean it makes any sense. For example consider the following nonsense.

So what is a condition?
We will leave these sorts of considerations and assume our conditions make sense, whatever that means. The axiom of unions states that for any collection of sets, there is a set consisting of all the elements in each of the sets in the collection. Of course this is also open to further consideration. What is a collection? Maybe it would be better to say “set of sets” or, given a set whose elements are sets there exists a set whose elements consist of exactly those things which are elements of at least one of these sets. If S is such a set whose elements are sets,

signify this union.
Something is in the Cartesian product of a set or “family” of sets if it consists of a single thing taken from each set in the family. Thus

signifies the Cartesian product.
The Cartesian product is the set of choice functions, a choice function being a function which selects exactly one element of each set of S. You may think the axiom of choice, stating that the Cartesian product of a nonempty family of nonempty sets is nonempty, is innocuous but there was a time when many mathematicians were ready to throw it out because it implies things which are very hard to believe, things which never happen without the axiom of choice.
A is a subset of B, written A ⊆ B, if every element of A is also an element of B. This can also be written as B ⊇ A. A is a proper subset of B, written A ⊂ B or B ⊃ A if A is a subset of B but A is not equal to B,A≠B. A ∩ B denotes the intersection of the two sets, A and B and it means the set of elements of A which are also elements of B. The axiom of specification shows this is a set. The empty set is the set which has no elements in it, denoted as ∅. A ∪ B denotes the union of the two sets, A and B and it means the set of all elements which are in either of the sets. It is a set because of the axiom of unions.
The complement of a set, (the set of things which are not in the given set ) must be taken with respect to a given set called the universal set which is a set which contains the one whose complement is being taken. Thus, the complement of A, denoted as A^{C} ( or more precisely as X ∖ A) is a set obtained from using the axiom of specification to write

The symbol
Words such as “all” or “there exists” are called quantifiers and they must be understood relative to some given set. For example, the set of all integers larger than 3. Or there exists an integer larger than 7. Such statements have to do with a given set, in this case the integers. Failure to have a reference set when quantifiers are used turns out to be illogical even though such usage may be grammatically correct. Quantifiers are used often enough that there are symbols for them. The symbol ∀ is read as “for all” or “for every” and the symbol ∃ is read as “there exists”. Thus ∀∀∃∃ could mean for every upside down A there exists a backwards E.
DeMorgan’s laws are very useful in mathematics. Let S be a set of sets each of which is contained in some universal set, U. Then

and

These laws follow directly from the definitions. Also following directly from the definitions are:
Let S be a set of sets then

and: Let S be a set of sets show

Unfortunately, there is no single universal set which can be used for all sets. Here is why: Suppose there were. Call it S. Then you could consider A the set of all elements of S which are not elements of themselves, this from the axiom of specification. If A is an element of itself, then it fails to qualify for inclusion in A. Therefore, it must not be an element of itself. However, if this is so, it qualifies for inclusion in A so it is an element of itself and so this can’t be true either. Thus the most basic of conditions you could imagine, that of being an element of, is meaningless and so allowing such a set causes the whole theory to be meaningless. The solution is to not allow a universal set. As mentioned by Halmos in Naive set theory, “Nothing contains everything”. Always beware of statements involving quantifiers wherever they occur, even this one. This little observation described above is due to Bertrand Russell and is called Russell’s paradox.