Section 1.1 Statements, Symbolic Representation, and Tautologies
23
A B A 0 B
T
T
F
T
F
T
F
T
T
F
F
T
52. The binary connective T is called the Peirce arrow, named for American philosopher Charles Peirce (not
for the antique automobile). The truth table for T is given here. In Chapter 8 we will see that this truth table
represents the NOR gate
A
B A T B
T
T
F
T
F
F
F
T
F
F
F
T
53. Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of
two values, False or True. Three-valued logic allows a third value of Null or “unknown” (Section 5.3 discusses the implications of three-valued logic on databases). The truth tables for this three-valued system
follow.
A
B
A ` B
A
B
A ~ B
A
A′
T
T
T
T
T
T
T
F
T
F
F
T
F
T
F
T
T
N
N
T
N
T
N
N
F
T
F
F
T
T
F
F
F
F
F
F
F
N
F
F
N
N
N
T
N
N
T
T
N
F
F
N
F
N
N
N
N
N
N
N
a. Viewing N as “unknown”, explain why it is reasonable to define T ` N = N, F ~ N = N, and
N′ = N.
Suppose the statement, “Flight 237 is on time,” is true, the statement, “Runway conditions are icy,” is
false, and the truth value of the statement, “Flight 51 is on time,” is unknown. Find the truth values of the
following statements.
b. Runway conditions are not icy and flight 51 is on time.
c. Flight 51 is on time and flight 237 is not.
d. Flight 51 is not on time or runway conditions are not icy.
54. Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of
two values, F or T, which we can think of as 0 or 1. In fuzzy logic, or many-valued logic, statement letters
are assigned values in a range between 0 and 1 to reflect some “probability” to which they are false or
true. A statement letter with a truth value of 0.9 is “mostly true” or “has a high probability of being true”
while a statement letter with a truth value of 0.05 “has a very high probability of being false.” Fuzzy logic
Show that every compound wff is equivalent to a wff using only the
connective 0 . (Hint: Use Exercise 47 and find equivalent statements
for A ` B and A′ in terms of 0 .)
Show that every compound statement is equivalent to a statement
using only the connective T. (Hint: See Exercise 51.)
23
A B A 0 B
T
T
F
T
F
T
F
T
T
F
F
T
52. The binary connective T is called the Peirce arrow, named for American philosopher Charles Peirce (not
for the antique automobile). The truth table for T is given here. In Chapter 8 we will see that this truth table
represents the NOR gate
A
B A T B
T
T
F
T
F
F
F
T
F
F
F
T
53. Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of
two values, False or True. Three-valued logic allows a third value of Null or “unknown” (Section 5.3 discusses the implications of three-valued logic on databases). The truth tables for this three-valued system
follow.
A
B
A ` B
A
B
A ~ B
A
A′
T
T
T
T
T
T
T
F
T
F
F
T
F
T
F
T
T
N
N
T
N
T
N
N
F
T
F
F
T
T
F
F
F
F
F
F
F
N
F
F
N
N
N
T
N
N
T
T
N
F
F
N
F
N
N
N
N
N
N
N
a. Viewing N as “unknown”, explain why it is reasonable to define T ` N = N, F ~ N = N, and
N′ = N.
Suppose the statement, “Flight 237 is on time,” is true, the statement, “Runway conditions are icy,” is
false, and the truth value of the statement, “Flight 51 is on time,” is unknown. Find the truth values of the
following statements.
b. Runway conditions are not icy and flight 51 is on time.
c. Flight 51 is on time and flight 237 is not.
d. Flight 51 is not on time or runway conditions are not icy.
54. Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of
two values, F or T, which we can think of as 0 or 1. In fuzzy logic, or many-valued logic, statement letters
are assigned values in a range between 0 and 1 to reflect some “probability” to which they are false or
true. A statement letter with a truth value of 0.9 is “mostly true” or “has a high probability of being true”
while a statement letter with a truth value of 0.05 “has a very high probability of being false.” Fuzzy logic
Show that every compound wff is equivalent to a wff using only the
connective 0 . (Hint: Use Exercise 47 and find equivalent statements
for A ` B and A′ in terms of 0 .)
Show that every compound statement is equivalent to a statement
using only the connective T. (Hint: See Exercise 51.)