This document contains an assignment that is due on April 5th or 6th depending on the class. It consists of 8 questions involving logical expressions and formulas. The questions cover topics like identifying propositions, writing logical expressions using operators, simplifying formulas, constructing formulas from truth tables, proving equivalences, and using resolution to prove statements.

1. Assignment 2
Deadline: April 5 (Thursday Class), April 6(Friday Class)
Harshit Kumar
March 29, 2012

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))
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