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Mathematical Logic and Proving Techniques Notes

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SUBJECT: MATHEMATICAL LOGIC AND PROVING TECHNIQUES CHAPTER: PROVING TECHNIQUES METHOD 7: INDUCTION METHOD In mathematics, a statement is a sentence which is either true or false. Given two statement A and B: If A is true, then B is true where statement A is called as hypothesis and statement B is the conclusion. When the statement B containing the quantifier “for all”, we can use the induction method to show the statement is true. Example; “For all integer n, something happens”. How to use induction method? Below are the steps to use induction. Step 1: Verify that P(1) is true by doing the Left-Hand side and Right-Hand side calculation. Step 2: Assume that P(n) is true. Step 3: Show that P(n+1) is true. Step 4: Prove that P(n+1) is true. Step 5: Conclusion. To understand deeply how to use induction method, look at the examples below. Website: https://mathsedu728404731.wordpress.com/ Example 1: Prove by induction that 𝑛 ∑𝑖 = 𝑖=1 𝑛(𝑛 + 1) 2 for every integer 𝑛 ≥ 1. Answer: Step 1: Verify that P(1) is true. Do RHS and LHS at n = 1. P ( n) : n = n ( n +1) 2 LHS: RHS: P (1) = 1 P (1) = 1(1+1) 2 2 = 2 =1 Since LHS = RHS, thus, P(1) is true. Step 2: Assume that P(n) is true. Write back the statement P(n). P ( n) : 1 + 2 + 3 + ... + n = n ( n+1) 2 Step 3: To show that P(n + 1) is true, substitute n = n+1 into the statement P(n). ( n +1)(( n +1)+1) 2 ( n +1)( n + 2) = 2 P ( n + 1) : 1 + 2 + 3 + ... + n + ( n + 1) = (n+1) from Step ...
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