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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.
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.
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.
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 .
Thus if any N horses are all the same colour, any N + 1 horses are the same colour. This is clearly true for N = 1 (i.e., one horse is a group where all the horses are the same colour). Thus, by induction, N horses are the same colour for any positive integer N, and so all horses are the same colour. The fallacy in this proof arises in line 3.
First, 2 is prime. Then, by strong induction, assume this is true for all numbers greater than 1 and less than n. If n is prime, there is nothing more to prove. Otherwise, there are integers a and b, where n = a b, and 1 < a ≤ b < n. By the induction hypothesis, a = p 1 p 2 ⋅⋅⋅ p j and b = q 1 q 2 ⋅⋅⋅ q k are products of primes.
Direct proof methods include proof by exhaustion and proof by induction. ... Since 2k 2 + 2k is an integer, n 2 is also odd. ∎ References Sources ...
Uniqueness: no positive integer n has two different Zeckendorf representations. The first part of Zeckendorf's theorem (existence) can be proven by induction. For n = 1, 2, 3 it is clearly true (as these are Fibonacci numbers), for n = 4 we have 4 = 3 + 1. If n is a Fibonacci number then there is