CS 173: Skills list for "Examlet A"

• Math prerequisites
  • Algebraic manipulations with
    • equations
    • inequalities
    • fractions
    • absolute value
    • squares and square roots
    • i (square root of -1)
  • 2nd order polynomials: solving, factoring, finding roots. Easy cases only: don't worry if you don't remember the quadratic formula.
  • Basic rules for manipulating exponents and logs
  • Defining and composing numerical functions. E.g. if f(x) = x-6 and g(x) = 7x, then g(f(x)) = 7(x-6).
• Numbers and sets
  • Know what sets these symbols represent: R, N, Z, Z⁺, Q, C.
  • Notation for set membership, e.g. x ∈ Z
  • Know that 0 belongs to N but not Z⁺. (There's two conventions about what's in N. This is the one we are using this term.)
  • Know the notations [a,b] and (a,b) for closed and open intervals of the real line.
  • Know the definitions of the floor and ceiling functions, i.e. ⌈ x ⌉ and ⌊ x ⌋
• Propositional and predicate logic
  • Know the truth tables for basic logical operators, especially implies. Know that, unless there is specific indication otherwise, "or" means inclusive or.
  • Know the meaning of the universal and existential quantifier, shorthand notation, and basic terminology such as "scope" of a quantifier.
  • Translate between English and logical shorthand. But we realize that it's hard to pin down the exact meaning of some English sentences.
  • Know the distributive, commutative, and associative laws and that "p implies q" is equivalent to "(not p) or q".
  • Given a new, fairly simple, logical equivalence, figure out whether it's correct or not and explain why using a truth table or counter-example.
  • Identify non-statements (e.g. questions) and statements which are neither true not false, because they contain variables not bound by a quantifier.
  • Decide whether a complex statement is true, given information about the truth of the basic statements it's made out of.
  • Identify the hypothesis and the conclusion of an if/then statement.
  • Given a statement, give its negation.
  • Given an if/then statement, give its converse, and contrapositive. Know that the contrapositive is equivalent to the original statement, but the converse is not.
  • Simplify a negation or contrapositive by moving all negations onto individual propositions. This requires knowing certain key logical equivalences: double negation, DeMorgan's laws, and the rules for negating if/then statements and quantifiers.
     
• Number Theory
  • Know definitions, decide if simple statements involving them are true, understand the corresponding shorthand notation:
    • a divides b, b is a multiple of a,
    • x is odd, x is even
    • x is prime, x is composite
    • x is a perfect square
    • x is a rational number
    • x and y are relatively prime
  • Apply the definition of divides to zero and to negative numbers (these aren't really 'edge cases' - the same exact definition applies!). For example, explain why zero is an even number.
  • Prime numbers, factoring a number into primes. (Where we only consider numbers >= 2.)
  • Special cases for prime: 0 and 1 are not prime (primes must be greater than or equal to 2).
  • Know the definitions of gcd(a,b) and lcm(a,b) and be able to compute gcd(a,b) and lcm(a,b) for specific (smallish) values of a and b.
  • Write any (small) positive integer as a product of primes.
  • Know that sqrt(2) is not rational.
  • Know that there are infinitely many primes.
  • Be able to state the Fundamental Theorem of Arithmetic: Any integer >= 2 can be written as the product of primes and each integer has only one prime factorization (except for the order in which you write the factors).
  • Define what it means for x and y to be congruent mod k
  • Be able to state the "Division Algorithm" theorem.
  • Compute the gcd of two larger numbers using the Euclidean algorithm, i.e. trace the execution of the algorithm.
  • Prove simple number theory claims using direct proof and the definitions of the terms involved. For example, if a divides b and a divides c, then a divides b+c.
• Logic and Proof techniques.
  • Write a simple direct proof, using familiar concepts, with good mathematical style. Make sure your statements are in logical order, starting with the given information and ending with what you needed to show.
  • Write a proof by cases
  • Convert a claim to its contrapositive and prove that using direct proof.
  • Know the following standard ways to approach parts of a proof:
    • prove a universal claim by working with a 'representative' object of the appropriate type (not a concrete example)
    • prove an if/then statement by assuming whatever's in the hypothesis and proving the conclusion
    • disprove a universal claim by giving a concrete counterexample
    • prove an existential claim by giving specific values that make the claim true