MathInduction - MATHEMATICAL INDUCTION Weak Mathematical...

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MATHEMATICAL INDUCTION “Weak” Mathematical Induction The principle of “weak” mathematical induction (“weak induction”, for short) builds upon our most fundamental intuition about the structure of the natural numbers: crudely put, that they are the result of starting with 0 and adding one, forever. Thus, if 0 has a certain property and if this prop- erty is also “inductive”, that is, roughly, if it is “passed on” from number to number through the add one relation, it will follow that every natural number has the property. This, in effect, is exactly the principle of weak induction; a bit more formally put: ( Weak Induction ) For any property ϕ of natural numbers: if 0 has ϕ and if, for any n N , from the assumption that n has ϕ it follows than n + 1 has ϕ , then all natural numbers have ϕ . Let’s illustrate how the principle is used with a simple example. Theorem. For every natural number n N , 0 + 1 + ... + n = n ( n + 1 ) 2 . Proof : First we establish the base case , where we show that 0 has the property in question — in this case the property being a natural number n such that 0 + 1 + ... + n = n ( n + 1 ) 2 . And it is obvious that it does, as, when n = 0, 0 + ... + n is just 0 and so we have: (0.1) 0 = 0 2 = 0 ( 1 ) 2 = 0 ( 0 + 1 ) 2 Now we prove the inductive case . This is where we show that if an arbitrary number n has the property in question, then so does its successor n + 1 . So assume that indeed the property is true of an arbitrary number n , that is, that (0.2) 0 + 1 + ... + n = n ( n + 1 ) 2 This assumption is called the induction assumption or, more commonly, the induction hypothesis , so-called because it is the assumption we make in or- der to establish the inductive nature of our property. Given the induction
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MATHEMATICAL INDUCTION 2 hypothesis (0.2), we need to show: (0.3) 0 + 1 + ... + n + ( n + 1 ) = ( n + 1 )(( n + 1 ) + 1 ) 2 . But (0.3) follows from (0.2) almost immediately. For by (0.2), we have (0.4) 0 + 1 + ... + n + ( n + 1 ) = n ( n + 1 ) 2 + ( n + 1 ) . And by a series of simple inferences we have a chain of identies establishing that n + 1 has the property in question: n ( n + 1 ) 2 + ( n + 1 ) = n ( n + 1 ) 2 + 2 ( n + 1 ) 2 (0.5) = n ( n + 1 ) + 2 ( n + 1 ) 2 (0.6) = ( n + 1 )( n + 2 ) 2 (0.7) = ( n + 1 )(( n + 1 ) + 1 ) 2 . (0.8) Therefore, since we have shown that 0 has the property in question, and that, if an arbitrary number n has it, so does its successor n + 1, we conclude by the principle of weak induction that all numbers have the property in question. EXERCISE 1 : Prove by weak induction that 1 2 + 2 2 + ...
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