Naive Set Theory - Sets, Membership and Equality

Sets, Membership and Equality

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, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.

If x is a member of A, then it is also said that x belongs to A, or that x is in A. In this case, we write x ∈ A. (The symbol ∈ is a derivation from the Greek letter epsilon, "ε", introduced by Peano in 1888.) The symbol ∉ is sometimes used to write x ∉ A, meaning "x is not in A".

Two sets A and B are defined to be equal when they have precisely the same elements, that is, if every element of A is an element of B and every element of B is an element of A. (See axiom of extensionality.) Thus a set is completely determined by its elements; the description is immaterial. For example, the set with elements 2, 3, and 5 is equal to the set of all prime numbers less than 6. If the sets A and B are equal, this is denoted symbolically as A = B (as usual).

We also allow for an empty set, often denoted Ø and sometimes : a set without any members at all. Since a set is determined completely by its elements, there can only be one empty set. (See axiom of empty set.) Note that Ø ≠ {Ø}.