Bridge to Abstract Mathematics: Mathematical Proof and Structures

2.1 BASIC CONCEPTS OF THE PROPOSITIONAL CALCULUS 57

Figure 2.1 Truth tables for negation, conjunction, and disjunction.

EXAMPLE 3 Determine under what truth conditions the statement form for
the compound statement "Either St, sin xdx # 0 and d/dx(Zx) = ~2"~'
or p, sin x dx = 0 and in 6 = (In 2)(ln 3)" is true. Is the statement itself
true or false?

Solution Let p represent the statement "p, sin xdx = 0," so that
''P "_. sin x dx # 0 corresponds to - p. Let q symbolize "dld~(2~) =
x2"- '" and let r denote "ln 6 = (In 2)(ln 3)." The main connective in the
given compound statement is "either... or." The two statements joined
by this disjunction are themselves compound, each involving the con-
nective "and." Specifically, the latter two statements, in symbols, have the
form -p A q and p A r. To signify that v is the main connective, we
use (as in the algebra of numbers) parentheses around the expressions

• p A q and p A r, arriving finally at ( - p A q) v (p A r) as the symbolic
  form of the given statement.
  We must next construct a truth table for this statement form. The first
  three columns should be headed by p, q, and r, whereas the last column
  (farthest to the right) has at its head the statement form (-p A q) v (p A r)
  itself. Intermediate columns need to be provided for any compound
  statement forms that occur as components of the final statement form,
  in this case - p, -- p A q, and p A r. The number of rows is the number
  of possible truth combinations of thgcomponent statement forms p, q,
  and r. Based on Exercise 2, our table should have eight rows; based on
  the number of column headings we've noted, it requires seven columns,
  as shown in Figure 2.2.
  Note that the truth values in column 5 were obtained with reference
  to columns 2 and 4 (linked by conjunction); column 7 resulted from col-
  umns 5 and 6 (linked by disjunction). Finally, notice that the statement
  form is true in fpur of the eight cases, namely, those in which either p
  and r are both true (rows 1 and 2), or p is false and q is true (rows 3 and
  7). Only one of the eight cases, of course, corresponds to the situation
  of our original example, namely, row 6 (Why?). The original compound
  statement in this example is false. 0