Strong Induction
Strong induction proves integer-indexed claims by making all earlier established cases available when proving the next case.
Summary
Strong induction proves integer-indexed claims by making all earlier established cases available when proving the next case. This fits recursive algorithms whose calls use smaller inputs of different sizes.
For a proposition starting at an integer , first prove directly. Then, for an arbitrary integer , assume for every integer with , and prove . This establishes for every integer . Prove additional initial cases directly whenever the inductive argument does not cover them.
🧠 Key Concepts
- Strong induction
- Inductive hypothesis
- Base cases
- Recursive dependencies
- Ordinary induction
- Well ordering
- Least counterexample
- Circular reasoning
- Decreasing integer measure
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When proving by strong induction, what may the inductive hypothesis assume?
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Strong Induction
📘 Overview Strong induction proves an integer-indexed claim by assuming it holds at every earlier index when establishing the next case. It is especially suited to recursive algorithms whose calls use smaller inputs of varying sizes.
🧠 Key Idea To prove the next case, strong induction makes all previously established cases available, rather than only the immediately preceding case. Strong and ordinary induction have equal logical power.
⚔️ Core Details: - Setup: Let be a proposition about input size , with integer starting point
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Part of 🌱 BSCS Year 1 — Computing & College Foundations
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