Skip to main content
  1. Life
  2. Education & Schools
  3. General Education

Chapter 2 of "Naive Set Theory"

See also

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.

Advertisement

Life

  • Transgender cop
    A transgender police officer is stepping down from her position to run for office
    Video
    Political Office
  • Easter eggs
    Craft delicate, hand-painted eggs with flowers and other designs celebrating spring
    Camera
    Easter Eggs
  • Subway message
    Subway customer finds 'Big Mama' written on her order
    Video
    Subway Message
  • Working from home
    Working from home can be an exciting venture. Get tips to ensure productivity
    Camera
    Get Tips
  • Limes
    Rising cost of limes could be putting the squeeze on your favorite restaurant
    Expensive Limes
  • Pope Francis
    Religion: Pope Francis instructs how to fight against Satan
    Morning Mass

User login

Log in
Sign in with your email and password. Or reset your password.
Write for us
Interested in becoming an Examiner and sharing your experience and passion? We're always looking for quality writers. Find out more about Examiner.com and apply today!