CSE 20 DISCRETE MATH Prof. Shachar Lovett Clicker frequency: CA

Todays topics Proof by induction Section 3.6 in Jenkyns, Stephenson

Mathematical induction

“For all integers n >= a, P(n).” Base case - push first domino Inductive step – n th domino pushes the n+1 th

Example

Proof by induction template

A.1 B.2 C.n D.n+1 E.Other

Proof by induction template

For the inductive step, we want to prove that IF the theorem is true for some n >=[basis], THEN the theorem is true for n+1. How do we prove an implication p→q? A.Assume p, WTS ¬q (“p and not q”). B.Assume p, WTS q. C.Assume q, WTS p. D.Assume p→q, show it does not lead to contradiction.

Proof by induction template

A.The negation is true. B.The theorem is true for some integer k+1. C.The theorem is true for n+1. D.The theorem is true for some integer n>=1

Proof by induction template

Proof of inductive step (Isolation inductive case for n) (Using inductive assumption for n) (Simplification)

Proof by induction template

Mathematical induction P(a)P(a+1)P(a+2)P(a+3)P(a+4) …

Mathematical induction P(a)P(a+1)P(a+2)P(a+3)P(a+4) …

Mathematical induction P(a)P(a+1)P(a+2)P(a+3)P(a+4) …

Another example: induction with sets Theorem: if |A|=n then |P(A)|=2 n. Proof by induction on n. Base case: Inductive case: Assume… WTS… Proof…

Another example: induction with sets Theorem: if |A|=n then |P(A)|=2 n. Proof by induction on n. Base case: Inductive case: Assume… WTS… Proof… A.Theorem is true for all n. B.Thereom is true for n=0. C.Theorem is true for n>0. D.Theorem is true for n=1.

Another example: induction with sets

A.Theorem is true for some set B.Theorem is true for all sets C.Theorem is true for all sets of size n. D.Theorem is true for some set of size n.

Another example: induction with sets

A.Theorem is true for some set of size >n. B.Thereom is true for all sets of size >n. C.Theorem is true for all sets of size n+1. D.Theorem is true for some set of size n+1.

Another example: induction with sets

Proof of inductive step

Proof of inductive step (contd)

Next class More fun with induction Read section 3.6 in Jenkyns, Stephenson