Write each of the following statements in the form ‘if …. then’ :

(i) A rhombus is a square only if each of its angles measures 90⁰.

(ii) When a number is a multiple of 9, it is necessarily a multiple of 3.

(iii) You get a job implies that your credentials are good.

(iv) Atmospheric humidity increase only if it rains

(v) If a number is not a multiple of 3, then it is not a multiple of 6.

Let A = [2, 3, 5, 7]. Examine whether the statements given below are true or false.

(i) ∃ x ∈ A such that x + 3 > 9.

(ii) ∃ x ∈ A such that x is even.

(iii) ∃ x ∈ A such that x + 2 = 6.

(iv) ∀ x ∈ A, x is prime.

(v) ∀ x ∈ A, x + 2 < 10.

(vi) ∀ x ∈ A, x + 4 ≥ 11

Rewrite the following statement in five different ways conveying the same meaning.

If a given number is a multiple of 6, then it is a multiple of 3.

Write the converse and contrapositive of each of the following :

(i) If x is a prime number, then x is odd.

(ii) If a positive integer n is divisible by 9, then the sum of its digits is divisible by 9.

(iii) If the two lines are parallel, then they do not intersect in the same plane.

(iv) If the diagonal of a quadrilateral bisect each other, then it is a parallelogram.

(v) If A and B are subsets of X such that A ⊆ B, then (X – B) ⊆ (X – A)

(vi) If f(2) = 0, then f(x) is divisible by (x – 2).

(vii) If you were born in India, then you are a citizen of India.

(viii) If it rains, then I stay at home.

Given below are some pairs of statements. Combine each pair using if and only if :

(i) p : If a quadrilateral is equiangular, then it is a rectangle.

q : If a quadrilateral is a rectangle, then it is equiangular.

(ii) p : If the sum of the digits of a number is divisible by 3, then the number is divisible by 3.

q : If a number is divisible by 3, then the sum of its digits is divisible by 3.

(iii) p : A quadrilateral is a parallelogram if its diagonals bisect each other.

q : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

(iv) p : If f(a) = 0, then (x – a) is a factor of polynomial f(x).

q : If (x – a) is a factor of polynomial f(x), then f(a) = 0.

(v) p : If a square matrix A is invertible, then |A| is nonzero.

q : If A is a square matrix such that |A| is nonzero, then A is invertible.