**Question:**

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 $2 k=6$.

(xii) None of the students in this class has passed.

**Solution:**

(i) The negation of the given statement is:

It is false that every natural number is greater than 0.

(Or)

Every natural number is not greater than $0 .$

(Or)

There exists a natural number which is not greater than 0 .

(ii) The negation of the given statement is:

It is false that both the diagonals of a rectangle are equal.

(Or)

There exists at least one rectangle whose both the diagonals are not equal.

(iii) The negation of the given statement is:

It is false that the sum of 4 and 5 is 8 .

(Or)

The sum of 4 and 5 is not 8 .

(iv) The negation of the given statement is:

It is false that the number 6 is greater than 4 .

(Or)

The number 6 is not greater than 4 .

(v) The negation of the given statement is:

It is false that every natural number is an integer.

(Or)

Every natural number is not an integer.

(Or)

There exists at least one natural number which is not an integer.

(vi) The negation of the given statement is:

It is false that the number $-5$ is a rational number.

(Or)

The number $-5$ is not a rational number.

(vii) The negation of the given statement is:

It is false that all cats scratch.

(Or)

There exists a cat which does not scratch.

(viii) The negation of the given statement is:

It is false that there exists a rational number $x$ such that $x^{2}=3$.

(Or)

There does not exists a rational number $x$ such that $x^{2}=3$

(ix) The negation of the given statement is:

It is false that all students study mathematics at the elementary level.

(Or)

It is not the case that all students study mathematics at the elementary level.

(x) The negation of the given statement is:

It is false that every student has paid the fees.

(Or)

It is not the case that every student has paid the fees.

(Or)

There exists at least a student who does not pay the fees.

(xi) The negation of the given statement is:

It is false that there is some integer $k$ for which $2 k=6$.

(Or)

It is not the case there is some integer $k$ for which $2 k=6$

(xii) The negation of the given statement is:

It is false that none of the students in this class has passed.

(Or)

It is not the case that none of the students of this class has passed.