CPSC 625 – Homework 4
due: Monday, October 16, 2006
1. Determine whether the following sets of propositional formulas are valid, satisfiable (but
not valid), or unsatisfiable by giving a truth table for each. Be sure to label each table with
your conclusion (valid, etc.), and indicate which truth assignments are satisfying.
a. {P→(Q→P)}
b. {(P→(Q→R)) →((P→Q)→(P→R))}
c. {(P∧(P→Q)) →Q}
d. {(P∧(P→Q)) → ¬Q}
e. {((M→X)∧ ¬X)→ ¬M}
f. {A→C, B →C, A ∨B , ¬C}
g. {P→Q, Q →R, P ∨Q, ¬Q∨ ¬R}
h. {(A⊕B⊕C),¬(A⊕B),¬(A⊕C),¬(B⊕C)}
2. Derive a proof of the proposition Afrom the following set of sentences by rules of
inference. Recall that a proof contains a sequence of derivations labelled with the indexes of
the sentences they were derived from, along with an indication of the inference rule used.
{D∧E→A, F ∧C→D, ¬(F∧C→B), C ∨B→E}
3. You are the proprietor of Sammy’s Sport Shop. You have just received a shipment of three
boxes filled with tennis balls. One box contains only yellow tennis balls, one box contains
only white tennis balls, and one contains both yellow and white tennis balls. You would
like to stock the tennis balls in appropriate places on your shelves. Unfortunately, the boxes
have been labelled incorrectly; the manufacturer tells you that you have exactly one box of
each, but that each box is definitely labelled wrong.
incorrect: white yellow both
observations: &%
'$ &%
'$ &%
'$
yellow white yellow
Given the initial (incorrect) labelling of the boxes above, and the three observations, use
Propositional Logic to derive the correct labelling of the middle box. Begin by writing down
a knowledge base you will need (such as what observing a white ball drawn from box 2
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