Determinants

Deduction: Generalization of Specific Instance

Example : Rohit is a man & All men eat food, therefore, Rohit eats food.
Induction: Specific Instances to Generalization
Example : Rohit eats food. Vikash eats food. Rohit and Vikash are men. Then, All men eat food
Statement is true for n=1, n=k & n=k+1, then, the Statement is true for all natural numbers n.




Steps of Principle of Mathematical Induction:
Step 1: Let P(n) be a result or statement formulated in terms of n(given question).
Step 2: Prove that P(1) is true
Step 3: Assume that P(k) is true
Step 4: Using Step 3, prove that P(k+1) is true
Step 5: Thus P(1) is true and P(k+1) is true whenever P(k) is true.
Hence, by the Principle of Mathematical Induction, P(n) is true for all natural numbers n.

Example: Prove that 2n > n for all positive integers n
Solution:
Step 1: Let P(n): 2n > n
Step 2: When n =1, 21 >1. Hence P(1) is true.
Step 3: Assume that P(k) is true for any positive integer k, i.e., 2k > k ... (1)
Step 4: We shall now prove that P(k +1) is true whenever P(k) is true.
Multiplying both sides of (1) by 2, we get
2* 2k > 2*k
i.e., 2 k + 1 > 2k
or, 2 k + 1 > k + k
or, 2 k + 1 > k + 1 (since k>1)
Therefore, P(k + 1) is true when P(k) is true. Hence, by principle of mathematical induction, P(n) is true for
every positive integer n.