Ordinal numbers | Parity (mathematics)

Even and odd ordinals

In mathematics, even and odd ordinals extend the concept of parity from the natural numbers to the ordinal numbers. They are useful in some transfinite induction proofs. The literature contains a few equivalent definitions of the parity of an ordinal α: * Every limit ordinal (including 0) is even. The successor of an even ordinal is odd, and vice versa. * Let α = λ + n, where λ is a limit ordinal and n is a natural number. The parity of α is the parity of n. * Let n be the finite term of the Cantor normal form of α. The parity of α is the parity of n. * Let α = ωβ + n, where n is a natural number. The parity of α is the parity of n. * If α = 2β, then α is even. Otherwise α = 2β + 1 and α is odd. Unlike the case of even integers, one cannot go on to characterize even ordinals as ordinal numbers of the form β2 = β + β. Ordinal multiplication is not commutative, so in general 2β ≠ β2. In fact, the even ordinal ω + 4 cannot be expressed as β + β, and the ordinal number (ω + 3)2 = (ω + 3) + (ω + 3) = ω + (3 + ω) + 3 = ω + ω + 3 = ω2 + 3 is not even. A simple application of ordinal parity is the idempotence law for cardinal addition (given the well-ordering theorem). Given an infinite cardinal κ, or generally any limit ordinal κ, κ is order-isomorphic to both its subset of even ordinals and its subset of odd ordinals. Hence one has the cardinal sum κ + κ = κ. (Wikipedia).

Video thumbnail

Introduction to Odd and Even Functions

This video defines odd and even functions and provides 2 basic examples of odd and even functions. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com

From playlist Solving Polynomial Inequality

Video thumbnail

How to Determine If a Function is Odd, Even or Neither

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

Is the function even or odd? How you can tell

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

Steps to determine if a polynomial function is even or odd

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

Determine If a Function is Odd Even or Neither

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

A new basis theorem for ∑13 sets

Distinguished Visitor Lecture Series A new basis theorem for ∑13 sets W. Hugh Woodin Harvard University, USA and University of California, Berkeley, USA

From playlist Distinguished Visitors Lecture Series

Video thumbnail

Zorn's Lemma, The Well-Ordering Theorem, and Undefinability | Nathan Dalaklis

Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse Diagrams, and the Ordinals will come up in this video as tools to get a better view of where the proof of Zorn's lemma comes from. ***Corre

From playlist The First CHALKboard

Video thumbnail

Determining If a Function is Even or Odd

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

How To Count Past Infinity

Support Vsauce, your brain, Alzheimer's research, and other YouTube educators by joining THE CURIOSITY BOX: a seasonal delivery of viral science toys made by Vsauce! A portion of all proceeds goes to Alzheimer's research and our Inquisitive Fellowship, a program that gives money and resour

From playlist Knowledge

Video thumbnail

Zorn's Lemma, The Well-Ordering Theorem, and Undefinability (Version 2.0)

Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse Diagrams, and the Ordinals / set theory will come up in this video as tools to get a better view of where the "proof" of Zorn's lemma comes f

From playlist The New CHALKboard

Video thumbnail

AQA Core 3 9.06 Simpson's Rule

I explain where Simpson's Rule for numerical integration originates and I work through an example using it.

From playlist [OLD SPEC] TEACHING AQA CORE 3 (C3)

Video thumbnail

Statistical data analysis | Statistical Data Science | Part 1

In this course you will learn how to analyze data. #Statistic plays important role in terms of data analysis. Here you will get exposed to utilize and understand various statistical method to analyse data. The following topic has discussed in this course. - Central tendency (mean and me

From playlist Data Analysis

Video thumbnail

Q3) (ii) Core 3 OCR Maths January 2010.mp4

Powered by https://www.numerise.com/ Q3) (ii) Core 3 OCR Maths January 2010 Simple Simpson's Rule www.hegartymaths.com http://www.hegartymaths.com/

From playlist Core 3 OCR (not MEI) January 2010 Past Paper Solutions

Video thumbnail

Algebraically Determine If the Polynomial Function is Even or Odd

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

Algebraically Determine If a Function is Even or Odd

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

How to Determine If a Function is Even or Odd

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Video thumbnail

AQA A-Level Further Maths J1-03 Numerical Methods: Deriving Simpson’s Rule

https://www.buymeacoffee.com/TLMaths Navigate all of my videos at https://sites.google.com/site/tlmaths314/ Like my Facebook Page: https://www.facebook.com/TLMaths-1943955188961592/ to keep updated Follow me on Instagram here: https://www.instagram.com/tlmaths/ Many, MANY thanks to Dea

From playlist A-Level Further Maths J1: Numerical Methods

Video thumbnail

Determine When a Function is Even or Odd

👉 Learn how to determine if a function is even or odd. A function is even if the graph of the function is symmetrical about the y-axis, or a function is even if f(x) = f(-x). A function is odd if the graph of the function is symmetrical about the origin, or a function is odd if f(-x) = -f(

From playlist Is the Function Even or Odd | Polynomial

Related pages

Idempotence | Parity (mathematics) | Natural number | Mathematics | Limit ordinal | Mathematical proof | Integer | Transfinite induction | Well-ordering theorem | Successor ordinal | Ordinal number