# Difference between revisions of "FA19:Prelim 2 guide"

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have had homework problems on will be tested more heavily. The exam will be | have had homework problems on will be tested more heavily. The exam will be | ||

closed book, closed notes, and will be 90 minutes long. | closed book, closed notes, and will be 90 minutes long. | ||

+ | |||

+ | '''Update:''' Based on where we got in lecture on 10/25, I've decided not to include automata theory in the scope of the exam. You should ignore the sample questions having to do with automata. | ||

Here are some sample questions from past prelims: [[Media:fa19-prelim2-sample.pdf]] | Here are some sample questions from past prelims: [[Media:fa19-prelim2-sample.pdf]] | ||

(solutions coming soon: [[Media:fa19-prelim2-sample-sol.pdf]]). | (solutions coming soon: [[Media:fa19-prelim2-sample-sol.pdf]]). | ||

+ | |||

+ | Review session [[Media:fa19-p2rev-board.pdf| board image]] And [https://vod.video.cornell.edu/media/t/1_c63qphr8 video]. | ||

Here is a brief summary of what we've covered; for an exhaustive list see the [[Main Page]]: | Here is a brief summary of what we've covered; for an exhaustive list see the [[Main Page]]: | ||

+ | |||

* [[Equivalence relation]]s | * [[Equivalence relation]]s | ||

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*** [[modular number]], | *** [[modular number]], | ||

*** <math>[[ℤ_m]]</math>, | *** <math>[[ℤ_m]]</math>, | ||

− | *** [[unit]], | + | *** [[unit]], [[inverse]], <math>[[ℤ_m^*]]</math>, <math>[[φ(m)]]</math>, |

− | |||

− | |||

− | |||

** Key algorithms: | ** Key algorithms: | ||

*** finding the [[base b representation]], | *** finding the [[base b representation]], | ||

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*** [[Euler's theorem]], | *** [[Euler's theorem]], | ||

*** [[Claim:⟦k⟧_m is a unit if and only if gcd(k,m) = 1]] | *** [[Claim:⟦k⟧_m is a unit if and only if gcd(k,m) = 1]] | ||

+ | <strike> | ||

* [[Automata]] theory | * [[Automata]] theory | ||

** Be able to build [[DFA]]s for simple problems | ** Be able to build [[DFA]]s for simple problems | ||

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** Key results | ** Key results | ||

*** Know how to prove that an automaton is correct (by giving and proving a specification for <math>[[\hat{δ}]]</math>, see [[SP19:Lecture 23 Automata constructions|Lecture 23 Automata constructions]]) | *** Know how to prove that an automaton is correct (by giving and proving a specification for <math>[[\hat{δ}]]</math>, see [[SP19:Lecture 23 Automata constructions|Lecture 23 Automata constructions]]) | ||

+ | </strike> |

## Latest revision as of 09:17, 30 October 2019

Prelim 2 will cover all material covered in lecture and homework between 9/27 and 10/25. I will update this guide after Monday's lecture to reflect what we have covered and what is in scope. Material covered on prelim 1 will not be directly tested, although you will still need to know how to write proofs and work with the basic mathematical objects necessary for those topics. Material that you have had homework problems on will be tested more heavily. The exam will be closed book, closed notes, and will be 90 minutes long.

**Update:** Based on where we got in lecture on 10/25, I've decided not to include automata theory in the scope of the exam. You should ignore the sample questions having to do with automata.

Here are some sample questions from past prelims: Media:fa19-prelim2-sample.pdf (solutions coming soon: Media:fa19-prelim2-sample-sol.pdf).

Review session board image And video.

Here is a brief summary of what we've covered; for an exhaustive list see the Main Page:

- Equivalence relations
- Induction
- Be able to do proofs by weak induction, strong induction, and structural induction (structural induction will be covered on 10/23).
- Understand inductively defined sets and inductively defined functions, including BNF notation (this will be covered on 10/23)
- Be able to apply the algorithms contained in inductive proofs

- Number theory
- Key definitions:
- Key algorithms:
- finding the base b representation,
- Euclid's GCD algorithm,
- finding Bézout coefficients,
- finding modular inverses,
- fast exponentiation
- computing φ(p) where is prime
- computing φ(pq) where and are distinct primes,
- RSA encryption and decryption

- Key results:

~~
~~

- Automata theory
- Be able to build DFAs for simple problems
- Understand how to prove that a DFA is correct
- Key definitions:
- Key results
- Know how to prove that an automaton is correct (by giving and proving a specification for Lecture 23 Automata constructions) , see

~~
~~