lecture11 - MA1100 Lecture 11 Mathematical Induction Axiom...

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1 MA1100 Lecture 11 Mathematical Induction Axiom of Induction Principle of Mathematical Induction Base Case Inductive Step Chartrand: section 6.1, 6.2
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Lecture 11 2 Example 1 Cardinality of power set Let A be a finite set with n element. How many elements does P (A) have? 8 3 4 2 | P (A)| 2 1 |A| A = {a} P (A) = { « , {a} } A = {a, b} P (A) = { « , {a}, {b}, {a, b} } A = {a, b, c} P (A) = { « , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} } 2 n n
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Lecture 11 3 Example 2 Regions in a circle Place n points on a circle and connect each pair of points with a line. Count the number of distinct regions partitioned by the lines. 4 Number of regions 3 2 Number of points n
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Lecture 11 4 Axiom of Induction If T is a subset of N such that: 1. 1 œ T 2. for every k œ N , if k œ T, then k+1 œ T then T = N This is called the Axiom of Induction implication within the hypothesis T 1 k œ T k+1 œ T
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Lecture 11 5 Axiom of Induction If T is a subset of N such that: 1. 1 œ T 2. for every k œ N , if k œ T, then k+1 œ T then T = N Z does not have a smallest number Z Z Z Counter-example: T = {-1, 0, 1, 2, 3, …} Axiom of Induction is a property of N It does not hold for Z.
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Lecture 11 6 Well Ordering Principle The set N is well-ordered. Definition Let S be a non-empty subset of R. S is said to be well-ordered if every non-empty subset of S has a smallest element . Well-Ordering Principle It can be shown that Axiom of Induction is equivalent to Well-Ordering Principle .
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Lecture 11 7 Principle of Mathematical Induction Let P(n) be an open sentence such that: 1.P(1) is true 2.For all k œ N , if P(k) is true, then P(k+1) is true Then P(n) is true for all n œ N Theorem (Principle of Mathematical Induction) Proof Let T be the set: T = {n œ N | P(n) is true} Given P(1) is true So 1 œ T (1) Given: If P(k) is true, then P(k+1) is true So if k œ T, then k+1 œ T (2) By axiom of induction, T = N So P(n) is true for all n œ N So N = {n œ N | P(n) is true}
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Lecture 11 8 Principle of Mathematical Induction PMI can be used to prove ( " n œ N ) P(n) is true
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This note was uploaded on 04/15/2010 for the course MATHS MA1101R taught by Professor Vt during the Spring '10 term at National University of Singapore.

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lecture11 - MA1100 Lecture 11 Mathematical Induction Axiom...

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