FA19:Prelim 2 guide

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.

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

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

Equivalence relations
- Relations: relation, reflexive, symmetric, transitive, equivalence relation, equivalence class, A/R, well-defined functions from A/R.
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:
  - quotient, remainder, base b representation, divisibility
  - greatest common divisor
  - Bézout coefficient,
  - equivalence mod m,
  - well-defined function,
  - modular number,
  - ,
  - unit,
  - inverse,
  - ,
  - ,
- Key algorithms:
  - finding the base b representation,
  - Euclid's GCD algorithm,
  - finding Bézout coefficients,
  - finding modular inverses,
  - fast exponentiation
  - computing φ(p) where [math]p [/math] is prime
  - computing φ(pq) where [math]p [/math] and [math]q [/math] are distinct primes,
  - RSA encryption and decryption
- Key results:
  - Euclid's GCD algorithm computes the greatest common divisor,
  - modular addition and modular multiplication are well-defined,
  - Euler's theorem,
  - Claim:⟦k⟧_m is a unit if and only if gcd(k,m) = 1
Automata theory
- Be able to build DFAs for simple problems
- Understand how to prove that a DFA is correct
- Key definitions:
  - alphabet, , language, string,
  - DFA, state, transition function, accept, L(M), recognize, [math][[\hat{δ}]] [/math]
- Key results
  - Know how to prove that an automaton is correct (by giving and proving a specification for [math][[\hat{δ}]] [/math], see Lecture 23 Automata constructions)