2.2 TAUTOLOGY, EQUIVALENCE, THE CONDITIONAL, AND BICONDITIONAL 59
- Determine whether each of the following statements is true or false, based on
the truth or falsehood of the component simple statements. (Note: Unlike what
was done in Example 3, you need not construct complete truth tables. You should,
however, express each statement in symbolic form, as we did in the first paragraph
of the solution to Example 3.)
(a) It is not the case that 5 is not an odd integer.
(b) Either the derivative of a linear function is its slope or the moon is made
of green cheese.
(c) e = lim,,, (1 + l/x)" and l/e = lim,,, (1 - llx)"
(d) The sum of two even integers is even and it is not the case that the product
of two odd integers is odd.
(e) Either^2 #^5 and the sine function is an even function or it is not the case
that every real number has a multiplicative inverse, that is, reciprocal.
(f) It is not the case that April, September, and Wednesday are names of
months.
*(g) April is not the name of a month, and September is not the name of a month,
and Wednesday is not the name of a month.
(h) Either April, September, or Wednesday is not the name of a month.
(i) It is not the case that either April, September, or Wednesday is the name of
a month.
(j) It is not the case that either April and September are not names of a month
or Wednesday is the name of a month.
2.2 Tautology, Equivalence, the Conditional,
and Biconditional
The compound statement form (-p A q) v (p A r) of Example 3, Article 2.1,
turned out to be true under some truth conditions and false under others.
We occasionally refer to such a statement form as a contingency. Statement
forms that are always true or always false are of particular importance.
DEFINITION 1
A statement form that is true under all possible truth conditions for its com-
ponents is called a tautology. A statement form that is false under all possible
truth conditions for its components is called a contradiction.
EXAMPLE 1 Show that the statement forms p v -p and p A -p are, res-
pectively, a tautology and a contradiction.
Solution We demonstrate this by means of truth tables. Since only
one unknown is involved in each statement form, our table will require
only two rows. For the sake of compactness, we use a single table for
