Write the negation of each of the following statements:
(i) Every natural number is greater than 0.
(ii) Both the diagonals of a rectangle are equal.
(iii) The sum of 4 and 5 is 8.
(iv) The number 6 is greater than 4.
(v) Every natural number is an integer.
(vi) The number -5 is a rational number.
(vii) All cats scratch.
(viii) There exists a rational number x such that `x^(2) = 3.`
(ix) All students study mathematics at the elementary level.
(x) Every student has paid the fees.
(xi) There is some integer k for which 2k = 6.
(xii) None of the students of this class has passed.

Explanation

(i) There exists a natural number which is not greater than 0.
(ii) It is false that both the diagonals of a rectangle are equal.
(iii) It is false that the surn of 4 and 5 is 8.
(iv) The number 6 is not greater than 4.
(v) There exists a natural number which is not an integer.
(vi) The number -5 is not a rational number.
(vii) There exists a cat which does not scratch.
(viii) There does not exist a rational number x such that `x^(2) = 3.`
(ix) There exists a student who does not study mathematics at the elementary level.
(x) At least one student has not paid the fees.
(xi) For all integers `k, 2k ne 6.`
(xii) At least one student of this class has passed.