Mathematical induction is a powerful proof technique in Discrete Mathematics. It helps establish the truth of statements for all natural numbers by verifying a base case and showing that if one case holds, the next must also be true.
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State the property P(n) to be proved for all natural numbers n ≥ n₀
- Clearly define the property P(n) that you want to prove.
- Ensure that P(n) is a statement that can be evaluated as true or false.
- Specify the starting point n₀, which is the smallest natural number for which the property is claimed to hold.
- Use precise mathematical language to avoid ambiguity in the property being stated.
- Example: "P(n): The sum of the first n natural numbers is (n(n + 1))/2."
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Establish the base case: Prove P(n₀) is true
- Verify that the property holds for the initial value n₀.
- Provide a clear and logical proof for P(n₀) using direct calculation or reasoning.
- This step serves as the foundation for the inductive process.
- Ensure that the base case is not skipped, as it is crucial for the validity of the induction.
- Example: Show that P(1) is true by calculating (1(1 + 1))/2 = 1.
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State the inductive hypothesis: Assume P(k) is true for some arbitrary k ≥ n₀
- Assume that the property P(k) holds for an arbitrary natural number k, where k is greater than or equal to n₀.
- This assumption is a critical step that allows you to build upon it in the next step.
- Clearly articulate the inductive hypothesis to avoid confusion later in the proof.
- This step does not require proof; it is an assumption for the sake of argument.
- Example: Assume P(k): The sum of the first k natural numbers is (k(k + 1))/2.
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Prove the inductive step: Show that P(k) ⇒ P(k+1)
- Using the inductive hypothesis, demonstrate that if P(k) is true, then P(k + 1) must also be true.
- This often involves algebraic manipulation or logical reasoning to connect P(k) to P(k + 1).
- Clearly outline each step in the proof to ensure clarity and rigor.
- This step is essential for establishing the validity of the induction process.
- Example: Show that (k(k + 1))/2 + (k + 1) = ((k + 1)(k + 2))/2.
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Conclude that P(n) is true for all n ≥ n₀ by the principle of mathematical induction
- State that since the base case is true and the inductive step has been proven, P(n) holds for all natural numbers n ≥ n₀.
- Emphasize that this conclusion follows directly from the principle of mathematical induction.
- Reinforce the importance of the logical structure of the proof in establishing the truth of P(n).
- This conclusion solidifies the result and demonstrates the power of mathematical induction.
- Example: Conclude that the formula for the sum of the first n natural numbers is valid for all n ≥ 1.