**Rewrite the following statement with “if-then” in five different ways conveying the same meaning.**

**If a natural number is odd, then its square is also odd.**

Solution:

The given statement can be written in five different ways as follows.

(i) A natural number is odd implies that its square is odd.

(ii) A natural number is odd only if its square is odd.

(iii) For a natural number to be odd, its necessary that its square is odd.

(iv) For the square of a natural number to be odd, it is sufficient that the number is odd.

(v) If the square of a natural number is not odd, then the natural number is not odd.

**Write the contrapositive and converse of the following statements.**

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

**(ii) It the two lines are parallel, then they do not intersect in the same plane.**

**(iii) Something is cold implies that it has low temperature.**

**(iv) You cannot comprehend geometry if you do not know how to reason deductively.**

**(v) x is an even number implies that x is divisible by 4**

Solution:

(i) The contrapositive is as follows.

*If a number x is not odd, then x is not a prime number.*

The converse is as follows.

*If a number x is odd, then it is a prime number.*

(ii) The contrapositive is as follows.

*If two lines intersect in the same plane, then they are not parallel.*

The converse is as follows.

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

(iii) The contrapositive is as follows.

*If something does not have low temperature, then it is not cold.*

The converse is as follows.

*If something is at low temperature, then it is cold.*

(iv) The contrapositive is as follows.

*If you know how to reason deductively, then you can comprehend geometry.*

The converse is as follows.

*If you do not know how to reason deductively, then you cannot comprehend geometry.*

(v) The given statement can be written as: *If x is an even number, then x is divisible by 4.*

The contrapositive is as follows.

*If x is not divisible by 4, then x is not an even number.*

The converse is as follows.

*If x is divisible by 4, then x is an even number.*

**Write each of the following statement in the form “if-then”.**

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

**(ii) The Banana trees will bloom if it stays warm for a month.**

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

**(iv) To get A+ in the class, it is necessary that you do the exercises of the book.**

Solution:

(i) If you get a job, then your credentials are good.

(ii) If the Banana tree stays warm for a month, then it will bloom.

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

(iv) If you want to get an A+ in the class, then you do all the exercises of the book.

**Given statements in (a) and (b). Identify the statements given below as contrapositive or converse of each other.**

**(a) If you live in Delhi, then you have winter clothes.**

**(i) If you do not have winter clothes, then you do not live in Delhi.**

**(ii) If you have winter clothes, then you live in Delhi.**

**(b) If a quadrilateral is a parallelogram, then its diagonals bisect each other.**

**(i) If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral is not a parallelogram.**

**(ii) If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.**

Solution:

(a)

(i) Contrapositive of the given statement (a).

(ii) Converse of the given statement (a).

(b)

(i) Contrapositive of the given statement (b).

(ii) Converse of the given statement (b).