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lecture11(complete) - MA1100 Lecture 11 Mathematical...

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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 Announcement Hand in homework 2 (according to tutorial group). Write your name, matric number and tutorial group on the cover page of your homework scripts. This homework will be returned after the mid- term test . Return of clickers ( No.1 to 150 only) Mid-term test : Oct 2 (12.00-1.30pm) at MPSH1. More information during lecture break
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Lecture 11 3 Example 1 Cardinality of power set Let A be a finite set with n element. How many elements does P (A) have? |A| 1 2 3 | P (A)| 2 4 8 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} } n 2 n
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Lecture 11 4 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. 1 2 1 2 3 4 1 2 3 4 5 6 7 8 Number of points 2 3 4 Number of regions 2 4 8 n Assume no three lines intersect at the same point except on the circumference
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Lecture 11 5 5 6 Example 2 Number of points 2 3 4 Number of regions 2 4 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 27 22 23 24 25 26 28 29 30 31 16 31 Regions in a circle Assume no three lines intersect at the same point except on the circumference
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Lecture 11 6 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 hypothesis conclusion implication within the hypothesis T 1 œ T k+1 œ T N = 2 3 3 2
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Lecture 11 7 Axiom of Induction If T is a subset of such that: 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 does not hold Counter-example: T = {-1, 0, 1, 2, 3, …} a Axiom of Induction is a property of N It does not hold for Z.
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Lecture 11 8 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 . Which of the following is/are well-ordered ? A. [0, 1] B. { x œ Z | x > -2} C. A set with 100 real numbers.
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Lecture 11 9 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} inductive hypothesis
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