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Chapter 2 of "Naive Set Theory"

In this article, we will learn more about how to symbolize sets. Suppose we have a set that categorizes people according to their hair color. One of the elements of that set may be "x is blond." This will be true of some of the elements of the set A, which is the set of all people according to their hair color, and it will be true of at least one other.

In order to symbolize the presence of a blond person in the set of all people according to their hair color, we would write:

{x ∈ A : x is blond}

"x ∈ A" is an example of a "sentence." For our purposes, there are two kinds of sentences:

1) Sentences of belonging - such as x ∈ A.

2) Sentences of equality - such as A = B.

Let's look at what is known as the Axiom of Specification, also known by its German name, Aussonderungsaxiom:

Axiom of Specification - Halmos defines this axiom as follows: "To every set A and to every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds."

To put it simply: For every condition there is a set whose elements x of set A for which S(x) holds. Each condition represents a situation where an element x, which is a member of A, holds. B is the set for which such a condition holds.

By "condition" is simply meant "sentence." The "condition" or "sentence", as we saw before, is simply the statement on the left side of the colon. So "x ∈ A" in the formula {x ∈ A : x is blond} would be the formula's "condition" or "sentence." B = {x∈A: S(x)}.

In order to write a formula which denies that an element is a member of a set, we can write "x∉A." We can also write "x∈A" or "not (x∈A)."

Halmos, Paul. "Naive Set Theory." Litton Educational Publishing, 1960.