M2.1: Proof by Induction Flashcards
What is the goal of Mathematical Induction?
Prove a statement P(n) is true for all n≥n_0
The process involves a base case and an inductive step.
What is the first step in Mathematical Induction?
Base Case: Prove P(n_0) is true for the initial value (e.g., n_0 =1)
The base case serves as the foundation for the induction process.
What is the second step in Mathematical Induction?
Inductive Step: Assume P(k) is true for an arbitrary k≥n_0 and prove P(k+1)
This step relies on the inductive hypothesis.
What does the conclusion of Mathematical Induction state?
By the principle of mathematical induction, P(n) is true for all n≥n_0
This conclusion follows from the base case and inductive step.
What is the goal of Strong Induction?
Prove a statement P(n) is true for all n≥n_0
Similar to mathematical induction but uses a stronger assumption.
What is the first step in Strong Induction?
Base Case: Prove P(n_0) is true for the initial value
This establishes the starting point for the induction.
What is the second step in Strong Induction?
Inductive Step: Assume P(n_0), P(n_0 + 1),…, P(k) are all true and prove P(k+1)
This step requires a stronger assumption than in mathematical induction.
What does the conclusion of Strong Induction state?
By the principle of strong induction, P(n) is true for all n≥n_0
This reinforces the validity of the statement across all relevant n.
What is a key difference between Mathematical Induction and Strong Induction?
Mathematical Induction assumes only the immediate previous case P(k) to prove P(k+1)
Strong Induction assumes all previous cases P(n_0) to P(k) for proving P(k+1).
When should you use Mathematical Induction?
When each step only depends on the immediate previous step
This is effective for simpler cases where only one previous case is relevant.
When should you use Strong Induction?
When proving a step requires knowing multiple or all previous steps
Common in problems involving divisibility, recursive definitions, or multiple prior truths.
