### Mathematical Reasoning Miscellaneous Exercise

## Question:

Check the validity of the statements given below by the method given against it (i) p : The sum of an irrational number and a rational number is irrational (by contradiction method) (ii) q : If n is a real number with n>3 then n^2>9 (by contradiction method)

## Answer:

(i) p : The sum of an irrational number and a rational number is irrational (by contradiction method)

Assume that the sum of an irrational number and a rational number is rational. Let x be an irrational number and y be a rational number. Then, x + y = z, where z is a rational number. This contradicts the fact that the sum of an irrational number and a rational number is irrational. Therefore, the statement p is true.

(ii) q : If n is a real number with n>3 then n^2>9 (by contradiction method)

Assume that n is a real number with n>3 but n^2≤9. Then, n^2 - 9 ≤ 0 Therefore, n^2 - 9< 0 This contradicts the fact that n>3. Therefore, the statement q is true.

## Question:

Write the following statement in five different ways conveying the same meaning p : If triangle is equiangular then it is an obtuse angled triangle

## Answer:

- If a triangle is equiangular, then it is an obtuse angled triangle.
- A triangle which is equiangular is also an obtuse angled triangle.
- If a triangle is equiangular, its angles are obtuse.
- Equiangular triangles are obtuse angled triangles.
- Obtuse angled triangles are always equiangular.

## Question:

Write the negation of the following statements: i) p : For every positive real number x the number x−1 is also positive ii) q : All cats scratch iii) r : For every real number x, either x>1 or x<1 iv) s : There exist a number x such that 0<x<1

## Answer:

i) p : There exists a positive real number x such that x−1 is not positive.

ii) q : Not all cats scratch.

iii) r : For every real number x, neither x>1 nor x<1.

iv) s : There does not exist a number x such that 0<x<1.

## Question:

Write each of the statements in the form “if p then q” (i) p : It is necessary to have a password to log on to the server (ii) q : There is traffic jam whenever it rains (iii) r : You can access the website only if you pay a subscription fee

## Answer:

(i) If it is necessary to have a password to log on to the server then q.

(ii) If it rains then there is traffic jam.

(iii) If you pay a subscription fee then you can access the website.

## Question:

Given below are two statements p : 25 is a multiple of 5 q : 25 is a multiple of 8 Write the compound statements connecting these two statements with “And” and “Or” In both cases check the validity of the compound statement

## Answer:

Using ‘And’: p And q: 25 is a multiple of 5 and 25 is a multiple of 8. Validity: False

Using ‘Or’: p Or q: 25 is a multiple of 5 or 25 is a multiple of 8. Validity: True

## Question:

Re write each of the following statements in the form “p if and only if q” (i) p : If you watch television then your mind is free and if your mind is free then you watch television (ii) q : For you to get an A grade it is necessary and sufficient that you do all the homework regularly (iii) r : If a quadrilateral is equiangular then it is a rectangle and if a quadrilateral is a rectangle then it is equiangular

## Answer:

(i) You watch television if and only if your mind is free. (ii) You do all the homework regularly if and only if you get an A grade. (iii) A quadrilateral is equiangular if and only if it is a rectangle.

## Question:

State the converse and contrapositive of each of the following statements: (i) p : A positive integer is prime only if it has no divisors other than 1 and itself (ii) q : I go to a beach whenever it is a sunny day (iii) r : If it is hot outside then you feel thirsty

## Answer:

(i) Converse: If a positive integer has no divisors other than 1 and itself, then it is prime.

Contrapositive: If a positive integer is not prime, then it has divisors other than 1 and itself.

(ii) Converse: If I go to a beach, then it is a sunny day.

Contrapositive: If it is not a sunny day, then I do not go to a beach.

(iii) Converse: If you feel thirsty, then it is hot outside.

Contrapositive: If it is not hot outside, then you do not feel thirsty.