Phi 270
Fall 2013
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4.2.x. Exercises

1.

Use derivations to establish each of the claims of entailment and equivalence shown below. (Remember that claims of equivalence require derivations in both directions.)

a. A ∧ B ⊨ A ∨ B
b. A ∧ B ⊨ B ∨ C
c. A ∨ B, ¬ A ⊨ B
d. A ∨ (A ∧ B) ⊨ A
e. A ∨ B, ¬ (A ∧ C), ¬ (B ∧ C) ⊨ ¬ C
f. A ∧ (B ∨ C) ⊨ (A ∧ B) ∨ C
g. A ∨ B, C ⊨ (A ∧ C) ∨ (B ∧ C)
h. A ∨ B, ¬ A ∨ C ⊨ B ∨ C
i. A ≃ (A ∧ B) ∨ (A ∧ ¬ B)

2.

Use derivations to establish each of the claims of equivalence below.
a. A ∨ A ≃ A
b. A ∨ B ≃ B ∨ A
c. A ∨ (B ∨ C) ≃ (A ∨ B) ∨ C
d. A ∨ (B ∧ ¬ B) ≃ A
e. ¬ (A ∨ B) ≃ ¬ A ∧ ¬ B
f. ¬ (A ∧ B) ≃ ¬ A ∨ ¬ B
3.

Use derivations to check each of the claims below; if a derivation indicates that a claim fails, confirm a counterexample that lurks in an open gap.

a. A ∨ B, A ⊨ ¬ B
b. A ∨ (B ∧ C) ≃ (A ∨ B) ∧ C
c. ¬ (A ∨ B) ≃ ¬ A ∨ ¬ B

For more exercises, use the exercise machine.

Glen Helman 01 Aug 2013