Principle of Mathematical Induction
Certain results or statements are formulated in terms of n, where n is a positive integer. To prove such statements, principle of mathematical induction is applied.
If P(n) is a given statement involving the natural number n such that
(i) The statement is true for n = 1, i.e., P(1) is true, and
(ii) If the statement is true for n = k (where k is some positive integer), then the statement is also true for n = k + 1, i.e., truth of P(k) implies the truth of P (k + 1).
Then, P(n) is true for all natural numbers n.
The word ‘induction’ means the generalisation from particular cases or facts. The statement “P(1) is true, i.e., P(n) is true for n = 1”, is called the basic step. The statement “P(m + 1) is true, whenever P(m) is true”, is called the inductive step. It is a conditional property.
Deductive Reasoning: This approach is referred as the “top-down” approach because a person starts at the top with a very broad set of information and work his or her way down to a specific conclusion.
Inductive Reasoning: This approach is referred as the “bottom-up” approach because a person starts at the bottom with a very specific set of information and work his or her way up to reach to a general conclusion or theory.
Inductive Set: A set S is said to be inductive set if 1 belongs to the set. Also, successor of the number belongs to the set whenever that number belongs to the set. Natural number set is the smallest subset of R holding this property.