Propositional Logic Practie Problems

Propositional Logic (also called Sentential Logic or Propositional Calculus) is a branch of logic that deals with propositions—statements that are either true or false.

Question 1) Consider the following statements:

• P: Good mobile phones are not cheap.
• Q: Cheap mobile phones are not good.
• L: P implies Q
• M: Q implies P
• N: P is equivalent to Q

Which one of the following about L, M, and N is CORRECT?

(A) Only L is TRUE. 

(B) Only M is TRUE. 

(C) Only N is TRUE. 

(D) L, M, and N are TRUE.

Solution:

Let a and b be two proposition
a: Good Mobile phones.
b: Cheap Mobile Phones.

P and Q can be written in logic as
P: a-->~b
Q: b-->~a.

Truth Table
a b ~a ~b P Q
T T F F F F
T F F T T T
F T T F T T
F F T T T T

it clearly shows P and Q are equivalent.
so option D is Correct

Question 2) Which one of the following is not equivalent to p <-> q

(A) (\neg p \vee q)\wedge(p \vee \neg q )

(B) (\neg p \vee q)\wedge(q \rightarrow p )

(C) (\neg p \wedge q)\vee(p \wedge \neg q )

(D) (\neg p \wedge \neg q)\vee(p \wedge q )

Conjunction of p and q, denoted by p∧q, is the proposition ‘p and q.'. The conjunction p ∧ q is True, when both p and q is True. 

Disjunction of p and q, denoted by p∨q, is the proposition ‘p or q.'. The disjunction p∨q is False when both p and q is False. 

Logical Implication - It is a type of relationship between two statements or sentences. Denoted by ‘p → q.'. The conditional statement p → q is false when p is true and q is false and true otherwise. i.e., p → q = ¬p ∨ q

Bi-Condition A bi-conditional statement is a compound statement formed by combining two conditionals under “and.” Bi-conditionals are true when both statements have the exact same truth value. 

Solution: 

A biconditional is true when both propositions have the same truth value: p↔q≡(p∧q)∨(¬p∧¬q)

Option (D) matches this directly.

Option (A): (¬p∨q) ∧ (p∨¬q) is equivalent to (¬p∨q) ∧ (¬q∨p), which is p ↔ q.

Option (B): q→p is ¬q∨p, so (B) is the same as (A).

Option (C): (¬p∧q) ∨ (p∧¬q)

is true exactly when p and q have opposite truth values: p⊕q = ¬(p↔q)

Only option which is not equivalent to p↔q is option (C). So, option (C) is correct.

∧ q) which is Option (D)

Practice Problems

Question 1: Given:

• P: “It is raining” (r)
• Q: “The ground is wet” (w)

Check if the statement (r \rightarrow w) \wedge (w \rightarrow r) is logically equivalent to \neg r \vee w Justify with truth table

Question 2: Simplify (p∨q) ∧(¬p∨q) to an equivalent expression using logical laws.

Question 3: Let:

• P: "If I study, I will pass." (s→p)
• Q: "If I do not pass, then I did not study." (¬p→¬s)
• R: "If I pass, then I studied." (p→s)

Which of the following is true?
(A) P and Q are equivalent, but not R
(B) P and R are equivalent, but not Q
(C) All three are equivalent
(D) None are equivalent

Question 4: Given P: p→(q∨r) and Q: (p→q)∨(p→r). Are P and Q logically equivalent? Justify using a truth table.