Why Read This Book? - Index of

1.1 Introduction to Logic 17
example, consider that p ∧ q should certainly be viewed as having the same meaning
as q ∧ p. To build a truth table would produce identical columns for p ∧ q and
q ∧ p. This is the way we define logical equivalence.
Definition 1.1.8 Two statements are said to be logically equivalent if they have
precisely the same truth table values. If U and V are logically equivalent, we write
U ⇔ V .
Notice that the two statements in Exercise 1.1.6 are logically equivalent. To say
“p AND either q or r” has the same meaning to us as “p and q,ORp and r.”This is
a sort of distributive property, where ∧ distributes over ∨, just like multiplication
distributes over addition in the real numbers.
EXERCISE 1.1.9 Show that ∨ distributes over ∧ by showing that p ∨ (q ∧ r) is
logically equivalent to (p ∨ q) ∧ (q ∨ r).
Parts (a) and (b) of Exercise 1.1.7 show that (p ∨ q) ∨ r is logically equivalent to
p ∨ (q ∨ r). We say that ∨ has the associative property, and we may allow ourselves
the freedom to write p ∨ q ∨ r to mean either (p ∨ q) ∨ r or p ∨ (q ∨ r).
EXERCISE 1.1.10 Does ∧ have the associative property? Verify your answer
with a truth table.
EXERCISE 1.1.11 Construct a statement using only p, q, ∧, ∨, and ¬ that is
logically equivalent to p ˙∨ q. Demonstrate logical equivalence with a truth table.
EXERCISE 1.1.12 Below are two logical equivalences called DeMorgan’s laws
(a name you will want to remember). Verify these forms of DeMorgan’s laws with
truth tables.
(a) ¬(p ∧ q) ⇔¬p∨¬q (b) ¬(p ∨ q) ⇔¬p∧¬q With Exercises 1.1.7(a) and (c), 1.1.10, and 1.1.12(a), we can extend one form
of DeMorgan’s law by using the following manipulation of logical symbols:
¬(p ∧ q ∧ r) ⇔¬[(p ∧ q) ∧ r] ⇔¬(p ∧ q) ∨¬r
⇔ (¬p ∨¬q) ∨¬r ⇔¬p∨¬q∨¬r (1.7)
EXERCISE 1.1.13 Mimic the manipulation in (1.7) to show that ¬(p ∨ q ∨ r) is
logically equivalent to ¬p ∧¬q ∧¬r.
EXERCISE 1.1.14 Use DeMorgan’s laws and symbolic manipulation to show
that ¬[(p ∨ q) ∧ r] is logically equivalent to (¬p ∧¬q) ∨¬r.