write each of the following using ‘if and only if’ :

(i) In order to get A grade, it is necessary and sufficient that you do all the homework regularly.

(ii) If you watch television, then your mind is free, and if your mind is free, then you watch television.

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.

Let p : If x is an integer and x² is even, then x is even,

Using the method of contrapositive, prove that p is true.

Consider the statement :

q : For any real numbers a and b, a² = b²⇒ a = b

By giving a counter-example, prove that q is false.