Bridge to Abstract Mathematics: Mathematical Proof and Structures

64 LOGIC, PART I: THE PROPOSITIONAL CALCULUS Chapter 2

3. Let p represent the statement "I pass the test." Let q symbolize "I pass the course.''
   Let r stand for "I make the dean's list."
   (a) Express as a statement in English:
   0) P v - q (P -+ 4) A (4 -+ r)
   (id) - p -+ - q (iv) q~-r
   (v) q+P (vi) q -+ r
   (vii) - p 4 - r (viii) (p -+ r) A (r 4 p)
   Ox) - P A 4
   (b) Express symbolically:
   (i) Passing the test will put me on the dean's list.
   (ii) Either I pass the course or I don't make the dean's list.
   (iii) If I don't pass the course, I don't make the dean's list.
   (iv) In order to pass the course, I must pass the test.
   (v) I passed the course, but I didn't make the dean's list.
   (vi) Passing the test is tantamount to passing the course.


4. Determine whether each of the following statements is true or false, based on the
   truth or falsehood of the component simple statements: (As in Exercise 4, Article 2.1,
   first express each statement in symbolic form.)
   (a) If 5 is not an odd integer, then 8 is prime.
   (b) If e = lim,, , (1 + llx)", then In e = 1.
   *(c) Ifeither2#50r4+5=9,then52#25.
   (d) If the moon is made of green cheese, then the derivative of a linear function
   does not equal its slope.
   (e) The sine function is even if and only if the cosine function is odd.

• (f) If p, sin x dx = 0 and dldx(2") # x2"- ', then In^6 = (In 2)(ln 3).
  (g) p, sin x dx = 0 if and only if either d/dx(2xx) = x2"- ' or In 6 = (In 2)(ln 3).
  *(h) If p, sin xdx = 0, then In 6 = (In 2)(ln 3) implies that dldx(2") = x2"- '.

2.3 Theorems of the Propositional Calculus

The theorems of the propositional calculus are the tautologies. In an in-
formal approach such as the one we use "proofs" of theorems are the truth
tables by which a statement form is seen to be a tautology. The tautologies
of primary interest for mathematics are those whose main connective is
either the biconditional or the conditional, that is, the equivalences and the
implications.
If p and q are compound statement forms, then the statement form
p ++ q is a tautology if and only if p and q have the same truth values under
all possible truth conditions; that is, if and only if p and q are logically
equivalent. Thus we refer to any tautology having the biconditional as its
main connective as an
propositional calculus
I portant results of this

d

equivalence. We begin our study of theorems of the

by focusing on important equivalences. Several im-

type are suggested in the following example.