Lebanese American University
Fall 2020
Byblos
Discrete Structures I
Test #1
Date: 24/10/2020
Duration: 1h 30
Name:
ID:
1. (20 pts) Let ϕ be the following predicate logic formula:
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¬ ∀x P (x, z) −→ ∃t Q(x, y, t) ∨ ∀y R(y, x) −→ S(z, u)
(a) List free and bound occurrences of variables in ϕ.
F
(b) Give the parse tree of ϕ.
(c) Find the prenex form of ϕ.
2. (30 pts) We Consider the following three sentences:
• Alan likes kangaroos, and either Betty likes frogs or Carl likes hamsters
• Provided that Betty likes frogs, Alan doesn’t like kangaroos.
• Betty likes frogs assuming that Carl likes hamsters.
Consider the following propositional logic variables:
• A: Alan likes kangaroos.
LS
• B: Betty likes frogs.
• C: Carl likes hamsters.
(a) Write three propositional logic formulas that represent the three sentences above using
A, B and C and logical connectives.
(b) We denote by Γ the set of the three formulas of the preceding question. Use three
different methods (by contradiction, formal proof and resolution proof) to prove that
the three sentences above (that is, the three elements of Γ) cannot be simultaneously
true.
3. (14 pts) Express the following in predicate logic:
(a) Anyone who trusts a politician is crazy.
(b) Only intelligent students understand me.
1
−→
4. (30 pts) Find a resolution and a formal proof for the following:
(a) ⊢ [ (p −→ q) ∧ (¬p −→ r) ] −→ (q ∨ r)
(b) ⊢ [ p −→ (q ∨ r) ] ∨ [ (q ∧ r) −→ p ]
(c) {(p ∨ q) −→ r ; ¬r ∨ s ; ¬s} ⊢ p −→ t
5. (16 pts) Find using Karnaugh maps a DNF and a CNF for the formula ϕ which has the
following truth table:
q
r
s
φ
F
p
T
T
T
T
T
T
T
F
F
T
T
F
T
T
T
F
T
T
T
F
T
T
T
T
T
T
F
F
F
LS
T
T
F
T
F
T
T
F
F
T
F
F
F
T
T
T
F
T
F
T
T
F
T
T
F
F
T
F
F
F
F
F
T
F
F
F
F
F
T
F
T
F
F
F
T
F
F
F
F
F
F
2