Principle of Mathematical Induction (PMI) The Principle of Mathematical Induction (PMI) is a formal mathematical technique used to establish the truth of a...
Principle of Mathematical Induction (PMI)
The Principle of Mathematical Induction (PMI) is a formal mathematical technique used to establish the truth of a statement for all natural numbers. It involves a series of steps known as induction steps that aim to show that the statement holds true for all natural numbers.
Key Idea:
PMI says that if we can prove that the statement holds true for a base case (a specific natural number), and that the statement holds true for any succeeding case (a natural number that is greater than the base case), then the statement holds true for all natural numbers.
Induction Steps:
Base Case:
Show that the statement holds true for a base case, e.g., for n = 1.
This can be done through direct calculation or using a simple mathematical argument.
Suceeding Case:
Assume that the statement holds true for some natural number k.
This means we need to show that the statement holds true for k + 1.
This is achieved by applying the same mathematical reasoning used in the base case to the specific case of k + 1.
Induction Hypothesis:
Contradiction:
If the statement is false for some natural number, then it cannot hold true for all natural numbers.
Therefore, the hypothesis that the statement holds true for all natural numbers must be false.
Conclusion:
By satisfying the conditions of PMI, we have proven that the statement holds true for all natural numbers.
This means that the original statement is true for all natural numbers.
Example:
Let's consider the statement:
If n is a natural number, then n^2 is also a natural number.
Base Case:
For n = 1, n^2 = 1^2 = 1, which is a natural number.
Therefore, the statement holds true for n = 1.
Suceeding Case:
Assume the statement holds true for some n = k.
That means n^2 = (k + 1)^2 = k^2 + 2k + 1, which is also a natural number.
Therefore, the statement holds true for n = k + 1.
Conclusion: