A naive theory in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. The words and, or, if ... then, not, for some, for every are treated as in

In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people,

Given two sets A and B, A is a subset of B if every element of A is also an element of B. In particular, each set B is a subset of itself; a subset of B that is not equal to B is called a proper subset.
If A is a subset of B, then one can also say that B is a superset of A,

Given two sets A and B, their union is the set consisting of all objects which are elements of A or of B or of both (see axiom of union). It is denoted by A ∪ B.
The intersection of A and B is the set of all objects which are both in A and in B. It is denoted by A ∩ B.

The simplest way to describe a set is to list its elements between curly braces (known as defining a set extensionally). Thus {1, 2} denotes the set

In certain contexts, one may consider all sets under consideration as being subsets of some given universal set. For instance, when investigating properties of the real numbers R (and subsets of R), R may be taken as the universal set. A true universal set is not included in

Intuitively, an ordered pair is simply a collection of two objects such that one can be distinguished as the first element and the other as the second element, and having the fundamental