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Lec4.1-2

# Lec4.1-2 - L4.1-2 Lecture Notes Mathematical Induction It...

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L4.1-2 Lecture Notes: Mathematical Induction It is often the case that we want to consider the truth of statements such as (1) = n i 0 i = n(n+1)/2, n 0 This means that the sum from zero to any natural number is given by the formula. This is another instance of a hidden universal quantifier. We could state it more explicitly by saying for all natural numbers, = n i 0 i = n(n+1)/2 or 2200 n = n i 0 i = n(n+1)/2 (1) is clearly a proposition. It is either true or false. The variable n is bound by the universal quantifier. Let’s symbolize the formula in logic. Let P(n): = n i 0 i = n(n+1)/2 P is a predicate and P(n) is a propositional function just like the ones we studied in logic. Give it a value for n , and it gives you back a truth value, for example: P(5): = 5 0 i i = 5(5+1)/2 = 15 is True since 0+1+2+3+4+5=15. In a universe of natural numbers, we may restate (1) as 2200 nP(n). It is often the case that we want to prove such statements as (1), but so far we have no way of doing so. Proving (1) is difficult because an infinite number of formulas have to be proved, namely for n=0, 1, . . It turns out that there is a rule of inference that will do the job for us. We will prove that rule of inference below. For now you may accept the following rule of inference as valid. 1

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RULE OF INFERENCE . P(0) Premise 2200 n(P(n) P(n+1)) Premise ————————— 2200 nP(n) Conclusion If we can show the truth of the two premises (somehow), then we are allowed to say we have proved the conclusion. This rule of inference has the name mathematical induction . This is consistent with our habit of naming rules of inference that we use often, like modus ponens or universal instantiation . Proofs that use mathematical induction as a rule of inference are often called “inductive proofs.” Intuitively, if we prove P for the value 0, then the truth of the second premise says that P is true for the value 1. But if it is true for 1, the second premise says that it is true for 2, and so ad infinitum. P(0) is proved directly. This step is the basis. The second premise is proved by using universal instantiation, and then using the proof method for implications, namely assuming the antecedent and proving the consequent. When we combine these we get a statement like this: Assume that P(n) is true for an arbitrary n. Based on this assumption , show that P(n+1) is true. (At this point the reader is advised to review the proof methods for implications given in Lecture Notes 1.5-7, p 8) (The curious reader may ask why we don’t simply attempt to prove the conclusion 2200 nP(n) by proving P for an arbitrary n . The problem is we don’t know how to express an arbitrary natural number by itself. But in an implication we can work with an arbitrary n in relation to n+1 . That’s an easier concept to work with, and that’s the strength of this rule of inference. It’s important to note that the
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Lec4.1-2 - L4.1-2 Lecture Notes Mathematical Induction It...

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