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Strong vs Weak Mathematical Induction Proof Technique

These flashcards cover the distinction between weak (simple) and strong (complete) mathematical induction — a core proof technique in discrete mathematics that students encounter in proof-writing courses and CS theory. Both forms share the same base case structure, but the inductive hypothesis differs in a subtle and powerful way. These cards help you recognize which form to use, execute both correctly, and avoid common errors that cost points on proofs exams and homework.

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5 Cards
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Front

What is the inductive hypothesis in weak induction?

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Back

In weak (simple) induction, the inductive hypothesis assumes P(k) is true for some arbitrary integer k ≥ the base case. You then prove P(k+1) is true using only P(k).

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Front

What is the inductive hypothesis in strong induction?

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Back

In strong (complete) induction, the inductive hypothesis assumes P(j) is true for all j where base ≤ j ≤ k. You then prove P(k+1) is true using any or all of P(base), ..., P(k).

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When should you use strong induction instead of weak induction?

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Use strong induction when proving P(k+1) requires knowledge of multiple prior cases, not just P(k). Common signals:

  • The proof involves recursive sequences (e.g., Fibonacci)
  • The problem reduces to two or more smaller sub-cases
  • P(k) alone is insufficient to prove P(k+1)
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Locked

Prove every n ≥ 2 is prime or a product of primes. Which induction form?

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What are the two required components of any induction proof?

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Frequently Asked Questions

What is the difference between strong and weak mathematical induction?

In weak induction, the inductive hypothesis assumes only P(k) is true. In strong induction, it assumes P(j) is true for all j from the base case up to k. Both are logically equivalent — every theorem provable by one can be proved by the other — but strong induction is far more convenient when P(k+1) depends on multiple prior values.

How do I know which form of induction to use?

Start with weak induction. If proving P(k+1) requires only P(k), weak induction works. If your proof needs P(k−1), P(k−2), or any case beyond just P(k), switch to strong induction. Recursive structures like Fibonacci sequences, binary trees, and prime factorization almost always require strong induction.

Do I need multiple base cases in strong induction?

Sometimes yes. If your inductive step uses both P(k) and P(k−1) to prove P(k+1), you must verify P(1) and P(2) as base cases. In general, verify as many base cases as your inductive step looks back — skipping extra base cases is one of the most common proof errors on exams.