Chapter 14: Mathematical Reasoning Miscellaneous Exercise

(i) The negation of statement p is given below.

There exists a positive real number x, such that x – 1 is not positive.

(ii) The negation of statement q is given below.

There exists a cat which does not scratch.

(iii) The negation of statement r is given below.

There exists a real number x, such that neither x > 1 nor x < 1.

(iv) The negation of statement s is given below.

There does not exist a number x, such that 0 < x < 1.

(i) Statement p can be written in the form ‘if then’ is as follows:

If a positive integer is prime, then it has no divisors other than 1 and itself.

The converse of the statement is given below.

If a positive integer has no divisors other than 1 and itself, then it is prime.

The contrapositive of the statement is given below.

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

(ii) The given statement can be written as follows:

If it is a sunny day, then I go to a beach.

The converse of the statement is given below.

If I go to a beach, then it is a sunny day.

The contrapositive of the statement is given below.

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

(iii) The converse of statement r is given below.

If you feel thirsty, then it is hot outside.

The contrapositive of statement r is given below.

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

(i) The statement p in the form ‘if then’ is as follows:

If you log on to the server, then you have a password.

(ii) The statement q in the form ‘if then’ is as follows:

If it rains, then there is a traffic jam.

(iii) The statement r in the form ‘if then’ is as follows:

If you can access the website, then you pay a subscription fee.

(i) You watch television if and only if your mind is free.

(ii) You get an A grade if and only if you do all the homework regularly.

(iii) A quadrilateral is equiangular if only if it is a rectangle.

The compound statement with ‘And’ is as follows:

25 is a multiple of 5 and 8.

This is a false statement because 25 is not a multiple of 8.

The compound statement with ‘Or’ is as follows:

25 is a multiple of 5 or 8.

This is a true statement because 25 is not a multiple of 8, but it is a multiple of 5.

(i) The given statement is as follows:

p: The sum of an irrational number and a rational number is irrational.

Let us assume that the statement p is false. That is,

The sum of an irrational number and a rational number is rational.

Hence,
where √a is irrational, and b, c, d, and e are integers.

∴ d / e – b / c = √a

But here, d / e – b / c is a rational number, and √a is an irrational number

This is a contradiction. Hence, our assumption is false.

∴ The sum of an irrational number and a rational number is rational.

Hence, the given statement is true.

(ii) The given statement q is as follows:

If n is a real number with n > 3, then n² > 9

Let us assume that n is a real number with n > 3, but n² > 9 is not true.

i.e., n² < 9

So, n > 3 and n is a real number.

By squaring both sides, we get

n² > (3)²

This implies that n² > 9, which is a contradiction since we have assumed that n² < 9

Therefore, the given statement is true, i.e., if n is a real number with n > 3, then n² > 9

The given statement can be written in five different ways; it is given below.

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.

(ii) A triangle is equiangular only if the triangle is an obtuse-angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled triangle.

(iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular.