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  2. Mathematical induction - Wikipedia

    en.wikipedia.org/wiki/Mathematical_induction

    Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.

  3. All horses are the same color - Wikipedia

    en.wikipedia.org/wiki/All_horses_are_the_same_color

    The argument is proof by induction. First, we establish a base case for one horse ( n = 1 {\displaystyle n=1} ). We then prove that if n {\displaystyle n} horses have the same color, then n + 1 {\displaystyle n+1} horses must also have the same color.

  4. Proofs involving the addition of natural numbers - Wikipedia

    en.wikipedia.org/wiki/Proofs_involving_the...

    We prove commutativity (a + b = b + a) by applying induction on the natural number b. First we prove the base cases b = 0 and b = S(0) = 1 (i.e. we prove that 0 and 1 commute with everything). The base case b = 0 follows immediately from the identity element property (0 is an additive identity), which has been proved above: a + 0 = a = 0 + a.

  5. Mathematical proof - Wikipedia

    en.wikipedia.org/wiki/Mathematical_proof

    For example, we can prove by induction that all positive integers of the form 2n − 1 are odd. Let P(n) represent "2n − 1 is odd": (i) For n = 1, 2n − 1 = 2(1) − 1 = 1, and 1 is odd, since it leaves a remainder of 1 when divided by 2. Thus P(1) is true.

  6. De Moivre's formula - Wikipedia

    en.wikipedia.org/wiki/De_Moivre's_formula

    The truth of de Moivre's theorem can be established by using mathematical induction for natural numbers, and extended to all integers from there. For an integer n, call the following statement S(n): (⁡ + ⁡) = ⁡ + ⁡. For n > 0, we proceed by mathematical induction.

  7. Structural induction - Wikipedia

    en.wikipedia.org/wiki/Structural_induction

    Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural numbers and can be further generalized to arbitrary Noetherian induction .

  8. Well-ordering principle - Wikipedia

    en.wikipedia.org/wiki/Well-ordering_principle

    Then show that for any counterexample there is a still smaller counterexample, producing a contradiction. This mode of argument is the contrapositive of proof by complete induction. It is known light-heartedly as the "minimal criminal" method [citation needed] and is similar in its nature to Fermat's method of "infinite descent".

  9. Fermat number - Wikipedia

    en.wikipedia.org/wiki/Fermat_number

    The Fermat numbers satisfy the following recurrence relations: = + = + for n ≥ 1, = + = for n2.Each of these relations can be proved by mathematical induction.From the second equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1.