Mathematics 43, section 1 -- Algebraic Structures

Assignment 1: Propositional and Predicate Logic

I. Construct a truth table for each of the following:
- A) (p => q) <=> ((p ^ (~ q)) => (~ p))
- B) ((p ^ q) => r ) <=> (p => (q => r))
- C) (p ^ q ^ r) => ((p v q) ^ (~ r))
II. For each of the following statements of the form p => q, identify p and q, give the converse, contrapositive, and inverse of the given statement, and determine whether each is true or false. Explain.
- A) If Mark McGwire hits more than 61 home runs this year, then he breaks Roger Maris's record.
- B) If my pet is a normal dog, then my pet has four legs.
- C) I satisfy the requirements for a mathematics major only if I pass this course.
- D) A triangle T being isoceles is sufficient for T being equilateral. (Note: in this sentence and the next, write p and q in good English as separate propositions -- do not just copy parts of this sentence! English grammar forces all sorts of changes if you use the ``is sufficient for'' and ``is necessary for'' constructions.)
- E) Having a valid US passport is necessary for traveling legally to Mexico.
III. For each of the following statements with quantifiers, give the symbolic form of the statement, the symbolic form of the negation, and write the negation as a sentence.
- A) For every natural number m, x^m is an integer.
- B) For all x in the real numbers, f(x) > 0.
- C) There exists a real number x such that for all real numbers y, f(x,y) = f(y,x).