Matematiksel tümevarım

Matematiksel tümevarım, arka arkaya düşen dominolara benzetilebilir.

Matematiksel tümevarım bir önermenin, genellikle tüm doğal sayılar için ya da bazen sonsuz bir sıranın tüm elemanları için, doğru olduğunu göstermek üzere kullanılan bir matematiksel kanıtlama yöntemidir. Matematiksel mantık ve bilgisayar bilimlerinde kullanılan daha genel bir tanıtlama biçimi değerlendirilebilen (hesaplanabilen) ifadelerin (dil için geçerli sözdizimlerinin) denk olduğunu gösterir. Buna yapısal tümevarım denir.

Matematiksel tümevarımın en basit ve en sık kullanılan şekli bir önermenin tüm doğal sayılar n için doğru olduğunu gösterir ve iki adımda gerçekleştirilir:

  1. Önermenin n = 0 için doğru olduğunu göstermek
  2. Önerme n = m için doğru ise aynı önermenin n = m + 1 için de doğru olacağını göstermek

Bu iki adımın neden yeterli olduğunu anlamak için domino etkisi örneğini göz önünde bulundurmak yeterli olacaktır. Baş başa dizilmiş olan bir domino taşları sırası var ve

  1. ilk domino taşını devirmek mümkün ise ve
  2. bir domino taşı devrildiğinde komşu taş da devriliyorsa, aynı şekilde dizilmiş olan ve sıranın devamı olan bütün domino taşlarının devrileceği sonucuna varılabilir.

Matematiksel tümevarım, kümeler için öngörülen İyi-sıralılık ilkesine denktir.

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