Why Read This Book? - Index of

22 Chapter 1 Language and Mathematics
Example 1.2.8 In Exercise 1.2.3, you constructed truth tables for q → p, ¬p →
¬q, and ¬q →¬p. The truth table values are displayed in Table 1.11. Notice that
p → q is logically equivalent to its contrapositive, and its converse and inverse are
logically equivalent.
p q ¬p ¬q p → q q → p ¬p →¬q ¬q →¬p
T T F F T T T T
T F F T F T T F
F T T F T F F T
F F T T T T T T
(1.11)
EXERCISE 1.2.9 Suppose x represents some specific, but unknown real number.
State the converse, inverse, and contrapositive of the following statement:
If x>1, then x 3 − x>0.
An important construct involving if-then is the statement that is true precisely
when p and q are either both true or both false. For example, if an integer is
divisible by 6, then it is divisible by both 2 and 3. Conversely, if an integer is
divisible by both 2 and 3, then it is divisible by 6. We say that an integer is divisible
by 6 if and only if it is divisible by both 2 and 3.
Definition 1.2.10 Given statements p and q, the statement p ↔ q (read “p if
and only if q,” and often written “p if q”) is defined to be true precisely when p
and q are either both true or both false.
EXERCISE 1.2.11 Construct a truth table for (p → q) ∧ (q → p) to show that
it is logically equivalent to p ↔ q.
EXERCISE 1.2.12 Let p, q, and r represent, respectively, the statements that
Penelope, Quentin, and Rhonda studied for the math test. Using p, q, r,¬, ∧, ∨, ˙∨,
→, and ↔, translate the following sentences into a symbolic logical construction.
(a) Rhonda studied for the test, but Penelope did not.
(b) If Rhonda studied for the test, then so did Penelope.
(c) If Rhonda did not study for the test, then Penelope and Quentin didn’t either.
(d) Quentin and Rhonda either both studied or both did not study.
(e) Neither Penelope nor Rhonda studied for the test.
(f) If Penelope did not study for the test, then neither did Rhonda.
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