CSE 20 DISCRETE MATH Prof Shachar Lovett http

Todays topics • Proof by induction • Section 3. 6 in Jenkyns, Stephenson

Mathematical induction “For all integers n >= a, P(n). ” • Base case -

Proof by induction template • A. 1 B. 2 C. n D. n+1 E.

Proof by induction template • For the inductive step, we want to prove that

Proof by induction template A. The negation is true. • B. The theorem is

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

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

Another example: induction with sets • A. Theorem is true for some set B.

Another example: induction with sets • A. Theorem is true for some set of

Next class • More fun with induction • Read section 3. 6 in Jenkyns,

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CSE 20 DISCRETE MATH Prof. Shachar Lovett http: //cseweb. ucsd. edu/classes/wi 15/cse 20 -a/ Clicker frequency: CA

Todays topics • Proof by induction • Section 3. 6 in Jenkyns, Stephenson

Mathematical induction •

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

Example •

Proof by induction template •

Proof by induction template • A. 1 B. 2 C. n D. n+1 E. Other

Proof by induction template •

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

Proof by induction template •

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) …

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: A. Theorem is true for all n. • Inductive case: B. Thereom is true for n=0. • Assume… C. Theorem is true for n>0. • WTS… D. Theorem is true for n=1. • Proof…

Another example: induction with sets •

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 •

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