EXERCISE 12.2

. 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 ∧¬ (iii) ¬ ∨ (iv) p →¬ (v) p ↔ .

Write each of the following sentences in symbolic form using statement variables p and q . (i) is not a prime number and all the angles of a triangle are equal. (ii) is a prime number or all the angles of a triangle are not equal (iii) is a prime number and all the angles of a triangle are equal (iv) is not a prime number . Determine the truth value of each of the following statements (i) If , then the milk is white.

(ii) China is in Europe or is an integer (iii) It is not true that or Earth is a planet (iv) is a prime number and all the sides of a rectangle are equal . Which one of the following sentences is a proposition? (i) (ii) What are you doing? (iii)  (iv) Peacock is our national bird (v) How tall this mountain is!

. Write the converse, inverse, and contrapositive of each of the following implication. (i) If x and y are numbers such that x , then x (ii) If a quadrilateral is a square then it is a rectangle . Construct the truth table for the following statements.

Prove that q ≡¬ →¬ . Show that p ® and q ® are not equivalent . Show that ¬ ↔ ≡ ↔¬ . Check whether the statement p ® ® ) is a tautology or a contradiction without using the truth table.

. Using truth table check whether the statements ¬ ∨¬ ∧ q and ¬ p are logically equivalent. . Prove p ) ≡ without using truth table.

. Prove that p →¬ ∨ ¬ ∨¬ ∨ ≡ ) using truth table. EXERCISE . Choose the correct or the most suitable answer from the given four alternatives.

. A binary operation on a set S is a function from