1. Fully parenthesize each of the following compound propositions. That is, add parentheses so that the structure is determined by the parentheses without the need for rules of precedence.

    1. p ∧ q ∨ r  (p ∧ q) ∨ r

    2. p ∧ q → r ∧ s   (p ∧ q) → (r ∧ s)

    3. p → q → r  (p → q) → r

  2. How many rows are in the truth table of compound proposition p → (q → (¬r → s)?

    There are 4 variables so there are 24 = 16 rows in the truth table.

  3. Construct a truth table of p → (q ∧ p).

    p q   p → (q ∧ p)
    F F   F T F F F
    F T   F T T F F
    T F   T F F F T
    T T   T T T T T


  4. Using a truth table, show that p → (q → p) is a tautology. (Show that it is true for all values of p and q.)

    p q   p → (q → p)
    F F   F T F T F
    F T   F T T F F
    T F   T T F T T
    T T   T T T T T


  5. What is the negation of each of the following propositions?

    1. Example: Jonathan has two cookies.   Jonathan does not have two cookies.

    2. There are 13 items in a baker's dozen.  There are not 13 items in a baker's dozen.

    3. Zelda drives more miles to school than Paola.  Zelda does not drive more miles to school than Paola.

    4. 2 + 4 = 5  2 + 4 ≠ 5

  6. Let p be proposition ``you get an A on the final exam,'' q be proposition ``you do every exercise in the book,'' and r be proposition ``you get an A in this class.'' Express each of the following English assertions in propositional logic using p, q and r.

    1. Example: If you do every exercise in the book then get an A in this class.  q → r This was incorrect in the assignment.

    2. You get an A in this class, but you do not do every exercise in the book.  r ∧ ¬q

    3. You get an A on the final, you do every exercise in the book and you get an A in this class.  p ∧ q ∧ r

    4. To get an A in this class, it is necessary for you to get an A on the final.  r → p.

      It might help if you think of this as saying p is implied by r. That is, if you get an A in this class, you must have got an A on the final.

      P is necessary for Q means Q → P.

    5. To get an A in this class, it is sufficient for you to get an A on the final.  p → r.

      P is sufficient for Q means P → Q.

    6. You will get an A in this class if and only if you either do every exercise in the book or you get an A on the final.  r ↔ q ∨ p