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CSCE 222 CSCE222 Homework 12 – Texas A&M

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CSCE 222 CSCE222 Homework 12 – Texas A&M

Homework 12

You must show your work in order to receive credit.

Checklist:

Did you…

1. abide by the Aggie Honor Code?

2. solve all problems?

3. start a new page for each problem?

4. show your work clearly?

5. type your solution?

6. submit a PDF to gradescope and correctly assign problems to pages?

Strong Induction Template:

LetP(n) := PREDICATE, when

nBASE CASE

Basis Step:P(BASE CASE1), . . . , P(BASE CASE m)

WORK TO SHOW BASE CASE(S)P(BASE CASE1), . . . , P(BASE CASE m) hold.

Inductive Step:

Description

CSCE 222 CSCE222 Homework 12 – Texas A&M

Homework 12

You must show your work in order to receive credit.

Checklist:

Did you…

1. abide by the Aggie Honor Code?

2. solve all problems?

3. start a new page for each problem?

4. show your work clearly?

5. type your solution?

6. submit a PDF to gradescope and correctly assign problems to pages?

Strong Induction Template:

LetP(n) := PREDICATE, when

nBASE CASE

Basis Step:P(BASE CASE1), . . . , P(BASE CASE m)

WORK TO SHOW BASE CASE(S)P(BASE CASE1), . . . , P(BASE CASE m) hold.

Inductive Step:(P(BASE CASE1), . . . , P(k)) =P(k+ 1)

AssumeP(i) forBASE CASE1ik some kBASE CASE m: STATEP(i)

ShowP(k+ 1) : STATEP(k+ 1)

WORK

USE INDUCTIVE HYPOTHESIS(by inductive hypothesis)

WORK TO SHOWP(k+ 1)(P(BASE CASE1), . . . , P(k)) =P(k+ 1) holds forkBASE CASE

.P(n) holds for allnBASE CASE1 by strong induction.

Structural Induction Template:

LetP(s) := PREDICATE, when s S

Basis Step:P(BASE CASE(S)OF RECURSIV E DEFINITION OF S)

WORK TO SHOW BASE CASE(S)

P(BASE CASE(S)) hold.

Recursive Step:(P(s1)∧ · · · ∧P(sk))=(P(t1)∧ · · ·P(tj)), wheret1, . . . , tjare constructed from

s1, . . . , sk

AssumeP(si) for 1ik, wheresiS: STATEP(i)

ShowP(tl) for 1lj, wheretlis constructed froms1, . . . , sk: STATEP(tl)

WORK

USE INDUCTIVE HYPOTHESIS(by inductive hypothesis)

WORK TO SHOW

CSCE 222 CSCE222 Homework 12 – Texas A&M

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