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Mathematical Induction: Proof Methods and Examples

This lecture covers the concept of mathematical induction and its application in proving statements involving objects. The lecture provides a step-by-step explanation of induction proof, including the induction basis and the induction step. It also includes examples of using mathematical induction to prove statements.

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Mathematical Induction: Proof Methods and Examples

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  1. MATH 310, FALL 2003(Combinatorial Problem Solving)Lecture 5,Wednesday, September 10

  2. A.2 Mathematical Induction • Let pn denote a statement involving n objects. • Induction proof of pn, for all n ¸ 0: • Initial step (Induction Basis): Verify that p0 is true. • Induction step: Show that if p0, p1, ..., pn-1 are true, then pn must be true. • Note: You have to prove p0. You also have to prove pn, but in the proof you may “pretend” that pn-1 or any other pk, k < n is true. • Note: Induction comes in various forms. For instance, sometimes the initial step involves some other small number, say, p1, or p3, ...

  3. Example 1 • Let sn = 1 + 2 + ... + n. (A) • Prove sn = n(n+1)/2. (B) • Proof by induction. • Initial step: • s1 = 1. (A). • s1 = 1(1+1)/2 = 1 (B). • Induction step: • sn = [1 + 2 + ... + (n-1)] + n = sn-1 + n. • Now assume sn-1 = (n-1)n/2 • sn = sn-1 + n = (n-1)n/2 + n = n(n+1)/2.

  4. 2.1 Euler Cylces • Homework (MATH 310#2W): • Read 2.2. Read Supplement I.(pp 46-48) Write down a list of all newly introduced terms (printed in boldface or italic) • Do Exercises A.2: 4,12,17,24 • Do Exercises 2.1: 2,10,12,17 • Volunteers: • ____________ • ____________ • Problem: 2.1:17. Challenge (up to 5 + 5 points): Do Exerecise 2.1: # 20 (requires computer programming).

  5. Multigraph • In a multigraph we may have: • Parallel edges • Loop edges (= loops). A B C D

  6. A B C D Königsberg Bridges • Great Swiss mathematician Leonhard Euler solved the problem of Seven Bridges of Königsberg by showing that it is impossible to walk across each bridge just once.

  7. Trails and Cycles • Path P = x1- x2 - ... – xn (all vertices distinct). • Circuit C = x1- x2 - ... – xn – x1 [a path with an extra edge (xn ,, x1 )]. • Trail T = x1- x2 - ... – xn (vertices may repeat but all edges are distinct). • Cycle E = x1- x2 - ... – xn – x1 [a trail with an extra edge (xn ,, x1 )].

  8. Euler Cycles and Trails • A cycle that uses every edge of a graph is called an Euler cycle (and visits every vertex). • A trail that uses every edge of a graph is called an Euler trail (and visits every vertex at least once).

  9. Theorem 1 (Euler, 1736) • An (undirected) multigraph has an Euler cycle if and only if: • it is connected and • has all vertices of even degree.

  10. Example 3: Routing Street Sweepers • Solid red edges represent a collection of blocks to be swept.

  11. Corollary • A multigraph has an Euler trail, but not an Euler cycle, if and only if • it is connected and • has exactly two vertices of odd degree.

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