Homework set 1: It's only logical

Due Tuesday, Feb 24.

Most of these are rather straightforward exercises, designed as sanity checks: to be sure we know and can apply the notation and basic logic concepts.

Ex i.j denotes exercise j at end of chapter i.
Ex i.j.k denotes exercise k at end of chapter i, section j.

  1. Ex 1.2(a)
  2. Ex 1.2(i)
  3. Ex 2.4
  4. Ex 2.5
  5. Ex 2.6(a)
  6. Ex 2.6(i)
  7. Ex 4.2.4
  8. Ex 4.2.5
  9. Ex 4.3.1(e), Also circle the free variables.
  10. 4.4.3.
The following discussion elaborates on a topic discussed in class. It is informative to think about this topic, but there is no required homework on it.

Observation: If a proposition contains 10 boolean variables, it's truth table will have over 1000 rows, if it contains 20 variables, the table will contain over a million rows. We can expect to encounter and want to verify statements about 10 or 20 variables in a program. Can we prove correctness of such statements in a less exaustive way? This thought leads us to consider deductive proof systems, as in chapter 2.

Meta-theorem: Proposition e is a tautology if and only if e = T is a theorem.
The proof of this meta-theorem is exercises 2.9 and 2.10.

Remember:

  • "Tautology" means "true in all states."
  • "is a theorem" means "result of a series of applications of the two rules of inference to the 12 laws (axioms)."

    In effect, exercise 2.9 is to prove: if e = T is a theorem then e is a tautology. We outlined this proof in class.

    In effect, exercise 2.10 is to prove: if e is a tautology then e = T is a theorem. It depends on exercise 2.8.

    Extra credit: do exercise 2.10, assuming 2.8 is correct.