RD Chapter 31- Mathematical Reasoning Ex-31.1 |
RD Chapter 31- Mathematical Reasoning Ex-31.3 |
RD Chapter 31- Mathematical Reasoning Ex-31.4 |
RD Chapter 31- Mathematical Reasoning Ex-31.5 |
RD Chapter 31- Mathematical Reasoning Ex-31.6 |

Write the negation of the following statement:

(i) Bangalore is the capital of Karnataka.

(ii) It rained on July 4, 2005.

(iii) Ravish is honest.

(iv) The earth is round.

(v) The sun is cold.

**Answer
1** :

(i) Bangalore is the capital of Karnataka.

The negation of the statement is:

It is false that “Bangalore is the capital of Karnataka.”

Or

“Bangalore is not the capital of Karnataka.”

(ii) It rained on July 4, 2005.

The negation of the statement is:

It is false that “It rained on July 4, 2005”.

Or

“It did not rain on July 4, 2005”.

(iii) Ravish is honest.

The negation of the statement is:

It is false that “Ravish is honest.”

Or

“Ravish is not honest.”

(iv) The earth is round.

The negation of the statement is:

It is false that “The earth is round.”

Or

“The earth is not round.”

(v) The sun is cold.

The negation of the statement is:

It is false that “The sun is cold.”

Or

“The sun is not cold.”

(i) All birds sing.

(ii) Some even integers are prime.

(iii) There is a complex number which is not a real number.

(iv) I will not go to school.

(v) Both the diagonals of a rectangle have the same length.

(vi) All policemen are thieves

**Answer
2** :

(i) All birds sing.

The negation of the statement is:

It is false that “All birds sing.”

Or

“All birds do not sing.”

(ii) Some even integers are prime.

The negation of the statement is:

It is false that “even integers are prime.”

Or

“Not every even integers is prime.”

(iii) There is a complex number which is not a real number.

The negation of the statement is:

It is false that “complex numbers are not a real number.”

Or

“All complex number are real numbers.”

(iv) I will not go to school.

The negation of the statement is:

“I will go to school.”

(v) Both the diagonals of a rectangle have the same length.

The negation of the statement is:

“There is at least one rectangle whose both diagonals do not have the same length.”

(vi) All policemen are thieves.

The negation of the statement is:

“No policemen are thief”.

Are the following pairs of statements are a negation of each other:

(i) The number x is not a rational number.

The number x is not an irrational number.

(ii) The number x is not a rational number.

The number x is an irrational number.

**Answer
3** :

(i) The number x is not a rational number.

“The number x is an irrational number.”

Since, the statement “The number x is not a rational number.” Is a negation of the first statement.

(ii) The number x is not a rational number.

“The number x is an irrational number.”

Since, the statement “The number x is a rational number.” Is not a negation of the first statement.

Write the negation of the following statements:

(i) p: For every positive real number x, the number (x – 1) is also positive.

(ii) q: For every real number x, either x > 1 or x < 1.

(iii) r: There exists a number x such that 0 < x < 1.

**Answer
4** :

(i) p : For every positive real number x, the number (x – 1) is also positive.

The negation of the statement:

p: For every positive real number x, the number (x – 1) is also positive.

is

~p: There exists a positive real number x, such that the number (x – 1) is not positive.

(ii) q: For every real number x, either x > 1 or x < 1.

The negation of the statement:

q: For every real number x, either x > 1 or x < 1.

is

~q: There exists a real number such that neither x>1 or x<1.

(iii) r: There exists a number x such that 0 < x < 1.

The negation of the statement:

r: There exists a number x such that 0 < x < 1.

is

~r: For every real number x, either x ≤ 0 or x ≥ 1.

Check whether the following pair of statements is a negation of each other. Give reasons for your answer.

(i) a + b = b + a is true for every real number a and b.

(ii) There exist real numbers a and b for which a + b = b + a.

**Answer
5** :

The negation of the statement:

p: a + b = b + a is a true for every real number a and b.

is

~p: There exist real numbers are ‘a’ and ‘b’ for which a+b ≠ b+a.

So, the given statement is not the negation of the first statement.

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