Multiple Mathematical Induction

Question

I need to prove a proposition in my research that requires induction on two variables? Is this possible using mathematical induction? I am seeing discussions of induction but with only one variable. Can you recommend books on how to do multiple induction so I can also write it on my reference list? Thanks to all.

Answer 1

In mathematical induction , The base case: prove that the statement holds for the first natural number n. Usually, n = 0 or n = 1, rarely, but sometimes conveniently, the base value of n may be taken as a larger number, or even as a negative number (the statement only holds at and above that threshold), because these extensions do not disturb the property of being a well-ordered set). The step case or inductive step: assume the statement holds for some natural number n, and prove that then the statement holds for n + 1. The hypothesis in the inductive step, that the statement holds for some n, is called the induction hypothesis or inductive hypothesis. To prove the inductive step, one assumes the induction hypothesis and then uses this assumption, involving n, to prove the statement for n + 1.

Whether n = 0 or n = 1 is taken as the standard base case depends on the preferred definition of the natural numbers. In the fields of combinatorics and mathematical logic it is common to consider 0 as a natural number.

but,in double variable mathematical induction.let,$p(m,n)$ is the required condition to be proved by induction. we show that it is true for $p(1,1)$ ,$p(1,n)$,$p(m,1)$ and next we will assume that it is true for $p(m_1,n_1)$ and show that t is also true for $p(m_1,n_1+1)$$p(m_1+1,n_1)$.this prove the $p(m,n)$ is true for all natural numbers