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Foundations of Computer Science

1. Use equivalences to transform the following wff into a CNF:

(A B) ¬ D → C

2. Give a formal proof for the following tautology by using the CP rule:

C (B → ¬ C) (B ¬ A) → ¬ A D

3. Give a formal proof for the following tautology by using the IP rule:

(¬ A ¬ D) (C → A) (B → D) → (¬ C ¬ B)

4. Find a countermodel for the wff x(p(x) → q(x)) x¬ q(x) → x¬ p(x).

5. Find a model for the wff x y p(x, y) x y ¬ p(x, y).

6. a) Determine L from the equation {^, b, baa}L = {^, aa, b, baa, baaaa}.

b) Find a grammar for the language {anbc 2n | n ≥ 0} = {b, abc2, a2bc 4, …, anbc 2n, …}.

7. Use the IP rule to prove that the following wff is valid:

x (p(x) → q(x)) ¬ x q(x) → ¬ x p(x)

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computer science

Formal Proof

tautology

CP rule

equivalences

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