### Mathematical Reasoning Exercise 04

## Question:

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

## Answer:

(a) (i) Contrapositive of (a) (ii) Converse of (a)

(b) (i) Contrapositive of (b) (ii) Converse of (b)

## Question:

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

## Answer:

(i) If your credentials are good then you get a job. (ii) If it stays warm for a month then the Banana trees will bloom. (iii) If the diagonals of a quadrilateral bisect each other then it is a parallelogram. (iv) If you do the exercises of the book then it is necessary to get A+ in the class.

## Question:

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

## Answer:

(i) Contrapositive: If x is not odd then x is not a prime number. Converse: If x is odd then x is a prime number.

(ii) Contrapositive: If the two lines intersect in the same plane then they are not parallel. Converse: If the two lines do not intersect in the same plane then they are parallel.

(iii) Contrapositive: If something does not have a low temperature then it is not cold. Converse: If something has a low temperature then it is cold.

(iv) Contrapositive: If you know how to reason deductively then you can comprehend geometry. Converse: If you cannot comprehend geometry then you do not know how to reason deductively.

(v) Contrapositive: If x is not divisible by 4 then x is not an even number. Converse: If x is divisible by 4 then x is an even number.

## Question:

Rewrite the following statement in five different ways conveying the same meaning. If a natural number is odd then its square is also odd.

## Answer:

- An odd natural number will always have an odd square.
- Every odd natural number has an odd square.
- The square of an odd natural number is always odd.
- Squaring an odd natural number always results in an odd number.
- If a natural number is odd, its square will be odd too.