Question 1: Write the negation of the following statements:

(i) Bangalore is the capital of Karnataka

It is not true that Bangalore is the capital of Karnataka. OR Bangalore is not the capital of Karnataka.

(ii) It rained on July 4,2005.

It is not true that it rained on July 4, 2005. OR It did not rain on July 4, 2005.

(iii) Ravish is honest.

It is not true that Ravish is honest. OR Ravish is not honest.

(iii) The sun is cold.

It is not true that the sun is cold. OR Sun is not cold.

(iv) The earth is round.

It is not true that earth is round. OR Earth is not round.

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Question 2: Negation of given statement:

(i) All birds sing.

Some birds do not sing. OR There exists a bird that does not sing.

(ii) Some even integers are prime.

Some integers are not prime. OR No even integer is prime.

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

All complex numbers are real numbers.

(iv) I will not go to school.

I will go to school.

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

Both  diagonals of a rectangle do no have the same length. OR Both diagonals of a rectangle have different lengths.

(vi) All policemen are thieves.

There exists a policeman that is not a thief. OR At least one policeman is not a thief.

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Question 3:

Are the following pairs of statements are 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.

(i) The statements in this pair are in negation of each other.

(ii) The statements in this pair are not in negation of each other because both statements are the same. Both the statement convey that $x$ is an irrational number.

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Question 4: 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 & either $x >1$ or  $x <1.$

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

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

Negation statement:

$\sim p:$ There exists a a positive real number x, such that the number $( x-1)$ is not positive.

(ii) $q:$ For every real number & either $x >1 \text{ or } x <1.$

Negation statement:

$\sim q :$ There exists a real number such that neither $x > 1 \text{ or } x < 1.$

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

Negation statement:

$\sim r:$ For every real number x, either $x\leq 0 \text{ or } x \geq 1.$

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Question 5:

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

(i) $a + b = b +a$ is true for every real number $a \text{ and } b.$

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

The given statements are not in negation of each other because the negation of $a + b = b +a$ is true for every real number $a \text{ and } b.$ is “There exists  real number $a \text{ and } b$  for which $a + b \neq b+ a.$