1. Let p : Jupiter is a planet and q : India is an island be any two simple statements. Give verbal sentence describing each of the following statements.
(i) ¬p (ii) p ∨ ¬q (iii) ¬p ∨ q (iv) p → ¬q (v) p ↔ q
2. Write each of the following sentences in symbolic form using statement variables p and q .
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number
3. Determine the truth value of each of the following statements
(i) If 6 + 2 = 5 , then the milk is white.
(ii) China is in Europe or √3 is an integer
(iii) It is not true that 5 + 5 = 9 or Earth is a planet
(iv) 11 is a prime number and all the sides of a rectangle are equal
4. Which one of the following sentences is a proposition?
(i) 4 + 7 = 12 (ii) What are you doing? (iii) 3n ≤ 81, n ∈ ℕ (iv) Peacock is our national bird (v) How tall this mountain is!
5. Write the converse, inverse, and contrapositive of each of the following implication.
(i) If x and y are numbers such that x = y , then x2 = y2
(ii) If a quadrilateral is a square then it is a rectangle
6. Construct the truth table for the following statements.
(i) ¬ p ∧ ¬q (ii) ¬( p ∧ ¬q) (iii) ( p ∨ q ) ¬q (iv) (¬ p → r) ∧ ( p ↔ q)
7. Verify whether the following compound propositions are tautologies or contradictions or Contingency
(i) ( p ∧ q ) ¬( p ∨ q)
(ii) ( ( p ∨ q) ¬p ) → q
(iii) ( p → q ) ↔ (¬ p → q)
(iv) ( ( p → q ) ∧ (q → r))→ ( p → r)
8. Show that (i) ¬( p ∧ q) ≡ ¬p ∨¬q (ii) ¬( p → q) ≡ p ∧¬q .
9. Prove that q → p ≡¬ p →¬q
10. Show that p → q and q → p are not equivalent
11. Show that ¬( p ↔ q) ≡ p ↔¬q
12. Check whether the statement p → ( q → p) is a tautology or a contradiction without using the truth table.
13. Using truth table check whether the statements¬( p ∨ q ) ∨ (¬p ∧ q) and ¬p are logically equivalent.
14. Prove p → ( q → r) ≡ ( p ∧ q) → r without using truth table.
15. Prove that p → (¬q ∨ r ) ≡ ¬p ∨ (¬q ∨ r) using truth table.
1. (i) ¬p : Jupiter is not a planet (ii) p ∧ ¬q : Jupiter is a planet and India is not an Island.
(iii) ¬ p ∨ q : Jupiter is not a planet or India is an Island.
(iv) p → ¬q : If Jupiter is a planet then India is not an Island.
(v) p ↔ q Jupiter is a planet if and only if India is an Island.
2. (i) ¬ p ∧ q (ii) p ∨ ¬q (iii) p ∧ q (iv) ¬p
3. (i) p → q is T (ii) p ∨ q is F (iii) ¬ p ∨ q is T (iv) p ∧ q is F
4. (i), (iii) and (iv) are propositions
5. (i) Converse: If x and y are numbers such that x2 = y2 then x = y .
Inverse: If x and y are numbers such that x ≠ y then x2 ≠ y2.
Contra positive: If x and y are numbers such that x2 ≠ y2 then x ≠ y .
(ii) Converse: If a quadrilateral is a rectangle then it is a square.
Inverse: If a quadrilateral is not a square then it is not a rectangle.
Contrapositive: If a quadrilateral is not a rectangle then it is not a square.
7. (i) Contradiction (ii) Tautology (iii) Contingency (iv) Tautology
12. p → ( q → p) is a Tautology.
13. Yes. The statements are logically equivalent.