Ordinal numbers | Parity (mathematics)
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).
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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
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From playlist The First CHALKboard
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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
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