Exercises 117
- Find the expression tree for the formula
p -- ((--p) -- q)
Evaluate the expression tree if proposition p is T and proposition q is E
- Find the expression tree for the formula
((p - --q) V q) -+ q
Evaluate the expression tree if proposition p is F and proposition q is T.
- Find the expression tree for the formula
((((-'(-'p)) A (-'q)) A r) V (((-(-,q)) A (-r)) A s) -• (s -+ p)
Evaluate the expression tree if proposition p is T, proposition q is T, proposition r is
F and proposition s is F
- Find the expression tree for the formula
((-,(p A q)) V (- (q A r))) A ((-'(p ++ (-(-'s)))) V ((r A s) V (-,q))).
Evaluate the expression tree if proposition p is F proposition q is T, proposition r is
F and proposition s is T
- Find the expression tree for the formula
-,(p A q) -• (-,p V --q)
Evaluate the expression tree for all possible pairs of truth values for p and q. Use these
evaluations to prove this formula is a tautology.
- For each of the following sets of propositions, identify a logically valid inference listed
in Table 2.6 that could be used to draw inferences from the formulas given. Identify
the rule of inference and what the inference rule implies.
(a) "If the sun is shining, then the courts will be open for play."
"If the courts are open for play, then we will play at 3 PM."
(b) "The sun is shining, or the courts are closed."
"The sun is not shining."
(c) "It is false that the sun is not shining."
(d) "If the courts are not open for play, then the sun is not shining."
(e) "If the sun is not shining, then the courts are not open for play."
"The courts are open for play."
(f) "If it is raining, then the courts are wet."
"If it is raining, then the courts are closed."
"If the courts are wet, then the courts are closed."
"The sun is shining."
9. Let 0 = "The home team is ahead." Let 7' = "The fans are happy." Let X = "The
visiting team is losing." For inference rules (a), (g), and (i) in Table 2.6, write out the
hypothesis and the conclusion for 0, V/, and X.
- Write the truth tables for the following formulas. Use the truth table to determine
whether any of these formulas is a tautology.
(a) ((p - q) A (q -- r)) - (p ÷ r)
(b) ((p - q) (qr) - ) (p -- r)
(c) ((p q) - r) -+(p -+(q -+r))
