2. Q1. Which of the following are proposition? (You answer should be T/F.)
1. y < 20
2. It is sunny.
3. open the closet.
4. 10 > 5
5. a2 + b2 = c2
Q2. Let H=”Kim is smart”, C=”Kim is cunning”, O=”Kim is intelligent”. Rewrite the following using the logical operators ∧, ∨, and ¬
1. Kim is smart and cunning but not intelligent.
2. Kim is either smart or cunning or both.
3. Kim is either smart or cunning but not both.
4. Kim is neither smart, cunning nor intelligent.
5. Kim is not both smart and cunning, but he is intelligent.
6. Kim is intelligent but not cunning nor smart.
Q3. Simplify the following logical formula
1. p ⊕ q
2. p → ¬q
3. (¬p ↔ ¬q) ∧ q
4. ¬(p → q)
Q4. Construct logical formula for the truth table below. After constructing the logical formula, simplify it.
a.
a b c f(a,b,c)
T T T T
T T F F
T F T F
T F F F
F T T F
F T F F
F F T T
F F F T
b.
a b c f(a,b,c)
T T T T
T T F F
T F T T
T F F T
F T T T
F T F F
F F T T
F F F F
Q5. Prove the equivalence of following logical formula? You answer should be T/F. You answer should also show why it is T/F. Show each step.
1. (P ∧ Q) ∨ R and P ∧ (Q ∨ R)
2. ¬(P ⊕ Q) and P ↔ Q
3. ¬(P ∨ Q ∨ R) and ¬P ∧ ¬Q ∧ ¬R
4. P ∧ (P ∨ Q) and P ∨ (P ∧ Q)
5. (P ∧ Q) ∨ (Q ∧ R) and Q ∨ (P ∧ R)
Q6. Simplify the following logical formula.
1. ¬((¬P ∧ (¬Q ∨ P )) ∨ ¬R).
2. (¬P ∧ ¬(P ∧ R)) ∨ ((Q ∨ (Q ∧ R)) ∧ (Q ∨ S)).
3. ((¬P ∧ Q) ∧ (Q ∧ R)) ∧ ¬Q.
4. ¬(¬Q ∧ ¬(¬Q ∨ S)) ∨ (Q ∧ (r → r)).
5. ¬P ∧ (P ∨ Q) ∨ (Q ∨ (P ∧ P )) ∧ (P ∨ ¬Q).
Q7. Simplify the following logical formula.
1. P → Q.
2. ¬Q → ¬P.
3. Q → P.
4. ¬P → ¬Q.
5. ¬(P → Q.)
Q8. Prove the following using resolution.
1. ((A → B) → C) → (¬C → A)
2. ¬(¬P ∨ ¬Q) → ((P → R) ∧ (Q → R))
1
