SP19:Prelim 2 guide

Prelim 2 will cover all material covered in lecture and homework between 3/1 and 4/1. 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: File:Sp19-prelim2-sample.pdf (solutions coming soon: File:Sp19-prelim2-sample-sol.pdf).

The slides from the review sessions:

session 1: automata; slides (video)
session 2: number theory slides (video)

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

Induction
- Be able to do proofs by weak induction, strong induction, and structural induction.
- Understand inductively defined sets and inductively defined functions, including BNF notation
- Be able to apply the algorithms contained in inductive proofs
Number theory
- Key definitions:
  - Some of this material was in scope for the first prelim: 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,
- 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]
  - Differences between NFA and DFA
  - Regular expressions, regular languages, the language of a regular expression
- Key results
  - Know and be able to apply the pumping lemma
  - 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)
  - Know and be able to apply the different parts of Kleene's theorem (converting DFA to NFA to RE, etc.)
  - Turing machines are not in scope.