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### 5.2 Strong Induction

• 1. Strong Induction and Well- Ordering Section 5.2 1
• 2. Strong Induction • Strong Induction: To prove that P(n) is true for all positive integers n, where P(n) is a propositional function, complete two steps: – Basis Step: Verify that the proposition P(1) is true. – Inductive Step: Show the conditional statement [P(1) ∧ P(2) ∧∙∙∙ ∧ P(k)] → P(k + 1) holds for all positive integers k. 2 Strong Induction is sometimes called the second principle of mathematical induction or complete induction.
• 3. Strong Induction and the Infinite Ladder 3 Strong induction tells us that we can reach all rungs if: 1. We can reach the first rung of the ladder. 2. For every integer k, if we can reach the first k rungs, then we can reach the (k + 1)st rung. To conclude that we can reach every rung by strong induction: • BASIS STEP: P(1) holds • INDUCTIVE STEP: Assume P(1) ∧ P(2) ∧∙∙∙ ∧ P(k) holds for an arbitrary integer k, and show that P(k + 1) must also hold. We will have then shown by strong induction that for every positive integer n, P(n) holds, i.e., we can reach the nth rung of the ladder.
• 4. Strong Induction and the Infinite Ladder 4 Strong induction tells us that we can reach all rungs if: 1. We can reach the first rung of the ladder. 2. For every integer k, if we can reach the first k rungs, then we can reach the (k + 1)st rung. To conclude that we can reach every rung by strong induction: • BASIS STEP: P(1) holds • INDUCTIVE STEP: Assume P(1) ∧ P(2) ∧∙∙∙ ∧ P(k) holds for an arbitrary integer k, and show that P(k + 1) must also hold. We will have then shown by strong induction that for every positive integer n, P(n) holds, i.e., we can reach the nth rung of the ladder.
• 5. Which Form of Induction Should Be Used? • We can always use strong induction instead of mathematical induction. But there is no reason to use it if it is simpler to use mathematical induction. • In fact, the principles of mathematical induction, strong induction, and the well-ordering property are all equivalent. • Sometimes it is clear how to proceed using one of the three methods, but not the other two. 5
• 6. Proof using Strong Induction Example: Prove that every amount of postage of 12 cents or more can be formed using just 4-cent and 5-cent stamps.
• 7. Proof using Strong Induction Solution: Let P(n) be the proposition that postage of n cents can be formed using 4- cent and 5-cent stamps. BASIS STEP: P(12), P(13), P(14), and P(15) hold. • P(12) uses three 4-cent stamps. • P(13) uses two 4-cent stamps and one 5-cent stamp. • P(14) uses one 4-cent stamp and two 5-cent stamps. • P(15) uses three 5-cent stamps. INDUCTIVE STEP: • The inductive hypothesis states that P(j) holds for 12 ≤ j ≤ k, where k ≥ 15. Assuming the inductive hypothesis, it can be shown that P(k + 1) holds. • Using the inductive hypothesis, P(k − 3) holds since k − 3 ≥ 12. To form postage of k + 1 cents, add a 4-cent stamp to the postage for k − 3 cents. Hence, P(n) holds for all n ≥ 12. 7
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