Exam 1 is in class, Thursday, October 5. See today’s notes for details an preparation advice. Thursday’s class may be an exam review - read directions on notes carefully, feel free to collude to hit optimal result. Class 11: Induction Practice cs2102: Discrete Mathematics | F17 uvacs2102.github.io David Evans | University of Virginia

Plan: All about Induction Induction Practice Induction in Practice Exam 1 is in class, Thursday, October 5. See today’s notes for details an preparation advice. Before 6:59pm Wednesday, send topics you would like to review (read directions on notes carefully, feel free to collude to hit optimal result).

What’s on Exam 1? MCS Chapters 1-5 Classes 1-11 and Notes

Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). = 2|5|

Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). To fit into exact induction principle: ∀ ∈ ℕ. ∷=

Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Prove ∀ ∈ ℕ. pow ℕ: = 2:

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Prove by Induction: sum of first positive integers is :(:;<) = Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1).

Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Prove that all non-empty finite subsets of ℕ have a minimum element. To fit into exact induction principle: ∀ ∈ ℕ. ∷=

Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). To fit into exact induction principle: ∀ ∈ ℕ. ∷= all subsets of ℕ of size + 1 have a minimum element. More naturally: ∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Prove that all non-empty finite subsets of ℕ have a minimum element.

Induction Principle To prove ∀ ∈ ℕ. : 1. Prove (0). 2. Prove ∀ ∈ ℕ. ⟹ ( + 1). Started last class: prove that all non-empty finite subsets of ℕ have a minimum element. To fit into exact induction principle: ∀ ∈ ℕ. ∷= all subsets of ℕ of size + 1 have a minimum element. More naturally: ∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ;. : 1. Prove (1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1). We can extend the induction principle to any well-ordered set with a “+ 1” operation that covers all the elements!

∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ;. : 1. Prove (1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1). 1. Prove 1 . “ ”

∀ ∈ ℕ;. ∷= all subsets of ℕ of size have a minimum element. Induction Principle+ To prove ∀ ∈ ℕ;. : 1. Prove (1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1). 2. Prove ∀ ∈ ℕ;. ⟹ ( + 1).“ ”

Take-Away Game Start with = 16 sticks Each turn: player must remove 1, 2, or 3 sticks Winner is player who takes the last stick Do you want to be Player 1 or Player 2?

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Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ;, ends.

Prove: Always Ends Theorem. A Take-Away game with any initial number of sticks, ∈ ℕ;, ends.

Charge Look at today’s notes (already posted) – Send requests for exam review by tomorrow PS5 Due Friday Exam 1 next Thursday (Oct 5)