Скачать или смотреть GI01 Mathematical Induction

#maths What is mathematical #induction? Induction is a mathematical proof technique for proving a sequence of (related) propositions. We explain the method and give 4 examples including ones from elementary #number theory and #graphtheory. In particular, we prove that if all edges of a simple connected graph are bridges then the number of edges is one less than the number of vertices. Subscribe ‪@Shahriari‬ for in depth math videos at the undergraduate level.
00:00 Introduction
00:11 The idea of mathematical induction
04:12 Motivating Problem: What is 1+3+5+...+1247?
04:38 Finding a pattern
07:10 A proof without words
07:52 Proof by Induction: the setup
09:57 Principle of Mathematical Induction (restated)
10:25 Example: Proof by Induction for the motivating problem
14:06 Second Example: Finding another pattern
15:01 Example 2: Pattern stated as a conjecture
15:27 Example 2 pattern proved by induction
17:19 Example 3 from number theory
17:56 Example 3 conjecture: 5^n + 2^n is divisible by 7 if n is odd
18:26 Example 3 conjecture proved by induction
20:48 Definition: What is a graph? (   • CG1 Graphs: Basic Vocabulary  )
21:54 Definition: What is a bridge? (   • CG3 Graph Theory: Walks, Paths, Cycles, an...  )
23:19 Theorem: If all edges of a simple connected graph are bridges then the number of edges is one less than the number of vertices (   • CG4 Graphs: Six Characterizations of Trees  )
23:31 Proof of Theorem by induction
Next video on Induction:    • GI02 Strong (Mathematical) Induction (and ...  

One of a series of lectures by Shahriar Shahriari on basic mathematical concepts used in undergraduate college mathematics.

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA USA
Shahriari is a 2015 winner of the Mathematical Association of America's Haimo Award for Distinguished Teaching of Mathematics, and six time winner of Pomona College's Wig teaching award.