maths.free › Combinatorics & Graph Theory › Induction › The Positive Integers are Well Ordered
The Positive Integers are Well Ordered
Most likely, you answered the questions posed in with an enthusiastic yes, in part because you wanted the shot at the money, but more concretely because it seems so natural.
The Positive Integers are Well Ordered
Most likely, you answered the questions posed in with an enthusiastic yes, in part because you wanted the shot at the money, but more concretely because it seems so natural. But you may be surprised to learn that this is really a much more complex subject than you might think at first. In , we discuss the development of the number systems starting from the Peano Postulates. Although we will not devote much space in this chapter to this topic, it is important to know that the positive integers come with some assembly required. In particular, the basic operations of addition and multiplication don't come for free; instead they have to be defined.
As a by-product of this development, we get the following fundamentally important property of the set \(\posints\) of positive integers:
An immediate consequence of the well ordered property is that the professor will indeed have to pay someone a dollareven if there are infinitely many students in the class.
Tagad tu Neviens kalkulators nenoslīgst šo, bet tā gabali ir komplējami. Izmēģiniet vienu zemāk vai ievadiet paši savu.
Bezmaksas konts pievieno piezīmes par katru nodarbību, ierakstu par to, ko esat pabeidzis, jūsu atrisinātās problēmas vienā vietā, un pasniedzēju, kuru varat jautāt par šo lapu. Pati matemātika ir pieejama ikvienam, kas ir vai nav parakstījies.
Pierakstīties PieteikšanāsŠeit izmantotie simboli
Piesitiet visiem simboliem, kas apzīmē pilnu definīciju, attēlu un ko katrs burts tajā nozīmē.
Jautājumi, ko cilvēki vaicā
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
Daļas šīs lapas ir pielāgotas no Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Šeit ir pārliecināts un vēlreiz izskaidrots; kļūdas ir mūsu.
Vairāk Combinatorics & Graph Theory
The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs