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}.
Read more about this topic: Naive Set Theory
Famous quotes containing the words membership and/or equality:
“The two real political parties in America are the Winners and the Losers. The people don’t acknowledge this. They claim membership in two imaginary parties, the Republicans and the Democrats, instead.”
—Kurt Vonnegut, Jr. (b. 1922)
“Prosperity or egalitarianism—you have to choose. I favor freedom—you never achieve real equality anyway: you simply sacrifice prosperity for an illusion.”
—Mario Vargas Llosa (b. 1936)