Mathematical Induction Introduction and basic problems CMSC 250
Mathematical Induction: Introduction and basic problems CMSC 250
Jason’s Tree of proof methods™ Statement S Universal Statement Universal Proof Indirect Direct Exhaustion Cases Defini tions … Contradiction Existential Proof Constructive Contraposition Non. Constructive Existential Statement
Jason’s Tree of proof methods™ Statement S Universal Statement Universal Proof Indirect Direct Exhaustion Cases Defini tions … Contradiction Existential Proof Constructive Contraposition Non. Constructive Existential Statement
Jason’s Tree of proof methods™ Statement S Universal Statement Universal Proof Indirect Direct Exhaustion Cases Defini tions … Contradiction Existential Proof Constructive Contraposition Induction Weak (Classical) Strong Structural Constructive Non. Constructive Existential Statement
Jason’s Tree of proof methods™ Statement S Universal Statement Universal Proof Indirect Direct Exhaustion Cases Defini tions … Contradiction Existential Proof Constructive Contraposition Induction Weak (Classical) Strong Structural Constructive Iffy Non. Constructive Existential Statement
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
The idea behind induction •
How we’ll make it work •
How we’ll make it work •
How we’ll make it work •
How we’ll make it work •
How we’ll make it work •
How we’ll make it work •
How we’ll make it work •
How we’ll make it work • (We fastforwarded here; time is of the essence!)
Our first inductive proof •
Inductive base • ?
Inductive Hypothesis •
Inductive step • This is our goal!
Inductive step • This is our goal!
Inductive step • This is our goal!
Inductive step, contd. •
Inductive step, contd. •
Inductive step, contd. •
Inductive step, contd. •
Inductive step, contd. •
Inductive step, contd. •
Inductive step, contd. •
Inductive step, contd. •
Here’s another •
Theorem expressed in quantifiers •
Inductive base •
Inductive hypothesis •
Inductive step •
Inductive Step (contd. ) •
Inductive Step (contd. ) •
Inductive Step (contd. ) •
Inductive Step (contd. ) •
Inductive step •
Inductive step •
Inductive step •
A note about the penny problem •
A note about the penny problem •
A note about the penny problem •
Here’s one with an inequality! •
Inductive Step… •
Inductive Step… •
Inductive Step… •
Inductive Step… •
Inductive Step… •
Inductive Step… •
Inductive Step… • Compressed Ind. Step
And one with more than 1 variable! •
And one with more than 1 variable! •
Proof •
Proof (contd. ) • By the I. H
Hey, wait a minute! •
Complete theorem •
And here’s four for you! •
- Slides: 80
