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Discrete_Mathematics 7 induction techniques.pptx
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- 2. Induction Mathematical induction isa prove technique that can be used to prove statements that assert that 𝑃(n) is true for all positive integers 𝑛,where 𝑃 (𝑛) is a propositional function.
- 3. Induction we complete 𝐭𝐰𝐨steps: Basis Step We verify that 𝑃 (1) is true. Inductive Step We show that the conditional statement 𝑃(𝑘) → 𝑃( 𝑘 + 1) is true for all positive integers 𝑘 we assume that 𝑃(𝑘) is true for an arbitrary positive integer 𝑘 and show that under this assumption, 𝑃(𝑘 + 1) must also be true. The assumption that 𝑃(𝑘) is true is called the inductive hypothesis (IH).
- 4. Induction EXAMPLE 1 Showthat if n is a positive integer, then 1) Basis Step: If 𝒏 = 𝟏. 𝑃 (1) is 𝐭𝐫𝐮𝐞, because This completes the basis step.
- 5. Induction 2) Inductive Step: Wefirst Assume that (Inductive Hypothesis (IH)) 𝑃 (𝑘) is true for the positive integer 𝑘, i. e. : 𝑃 (𝑘)="1 + 2 + 3 · · · + 𝑘 = ". We need to show that if 𝑃( 𝑘) is true, then 𝑃 (𝑘 + 1) is true. We add (𝒌 + 𝟏) to both sides of the equation in 𝑃 (𝑘) , we obtain 1 + 2 + 3 · · · + 𝑘 + (𝑘 + 1) IH =
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- 12. Strong Induction: Which canoften be used when we cannot easily prove a result using mathematical induction.
- 13. Strong Induction: EXAMPLE Showthat if n is an integer greater than 1, then n can be written as the product of primes.
- 14. Recursion The process ofdefining an object in terms of itself. Example 1: We use two steps to define a function with the set of nonnegative integers as its domain: 1) Basis Step: Specify the value of the function at the first point. 2) Recursive Step: Specifying how terms in the function are found from previous terms.
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- 16. Recursion Example: The sequence ofpowers of 2 is given by for 𝑛 = 0, 1, 2, … 1) Basis Step: Specify the value of the sequence at zero. 2) Recursive Step: Give a rule for finding a term of the sequence from the previous one. , for 𝑛 = 0, 1, 2,…
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