Bridge to Abstract Mathematics: Mathematical Proof and Structures

2.3 THEOREMS OF THE PROPOSITIONAL CALCULUS 73

Exercises

1. (a) In Exercise 3, Article 2.2, find any examples of pairs of statement forms
   [in (a)] or statements [in (b)] that are either equivalent to or negations of each
   other.
   (b) In Exercise 4, Article 2.2, write the negation of each statement, using parts
   (b) through (f) of Theorem 1 to avoid having negation as the main connective.
   Write the converse of all except (e) and (g).


2. (a) Prove, by using a truth table, five of the equivalences listed in Theorem 1.
   More specifically, choose one from (c) through (f), one from (g) through (I), one
   from (m) through (o), and two from (p) through (t).
   (b) Verify, by using a truth table, three of the implications in Theorem 2, choosing
   (g) and (j) and one from among (b), (i), and (k).


3. Determine which three of the implications in Theorem 2 are reversible, that is,
   have a converse that is also a tautology (recall remarks from the paragraph im-
   mediately following Theorem 2).


4. (a) Write the converse, contrapositive, and inverse of each of the following
   conditionals. (Your final answer in each case should not use the "not" connec-
   tive to modify a compound statement form. Use (b) through (f) of Theorem 1
   to simplify any such expressions that occur.)

(b) Fill in each of the blanks with one of the words original, converse, contra-

positive, or inverse.

(i) The converse of the converse of p -, q is the of p -, q.

(ii) The inverse of the contrapositive of p -, q is the of p -, q.

*(iii) The contrapositive of the converse of p + q is the of p + q.

(iv) The converse of the inverse of p -, q is the of p -, q.

(v) The inverse of the converse of p -, q is the of p -, q.

(vi) The original of the converse of p -+ q is the of p -, q.

*(vii) The contrapositive of the contrapositive of p -, q is the of

P -, 4.

(c) Give a concise description of the pattern emerging from the answers in (b),

governing the effect of two successive applications of the operations original,

contrapositive, converse, and inverse to a given statement form p -, q.

(d) Write the converse, inverse, contrapositive, and negation of the statements:

(i) If the economy improves, then I get a better job.

*(ii) If2<4and5+5= lO,thensin(n/3)=3.

(iii) If I finish my work, I play golf unless it rains.

(iv) Iff differentiable implies f is continuous, then f is continuous.

(v) Iff is defined at a, then the existence of lim,,, f (x) implies f is continuous

at a.

(vi) If (F, +, a) is a field, then (F, +) and (F - (01, .) are both abelian groups.