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**Exercise 14.3**

**1. For each of the following compound statements first identify the connecting words and then break it into component statements:**

**(i) All rational numbers are real and all real numbers are not complex.**

**(ii) Square of an integer is positive or negative.**

**(iii) The sand heats up quickly in the sun and does not cool down fast at night.**

**(iv) **** and **** are the roots of the equation **

**Ans. (i) **The component statement has the connecting word ‘and’ component statements are:

All rational numbers are real.

All real numbers are not complex.

**(ii) **The component statement has the connecting word ‘or’ component statements are:

Square of an integer is positive.

Square of an integer is negative.

**(iii) **The component statement has the connecting word ‘and’ component statements are:

The sand heats up quickly in the sun.

The sand does not cool down fast at night.

**(iv) **The component statement has the connecting word ‘and’ component statements are:

x = 2 is a root of the equation

y = 3 is a root of the equation

**2. Identify the quantifier in the following statements and write the negation of the statements:**

**(i) There exists a number which is equal to its square.**

**(ii) For every real number **** is less than **

**(iii) There exists a capital for every state in India.**

**Ans. (i) **Here the qualifier is ‘there exists’.

The negation of statement is: There exists a number which is not equal to its square.

**(ii) **Here the qualifier is ‘for every’.

The negation of statement is: For at least one real number is not less than

**(iii) **Here the qualifier is ‘there exists’.

The negation of statement is: There exists a state in India which does not have a capital.

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

**(i) **** is true for every real numbers **** and **

**(ii) There exists real numbers **** and **** for which ****.**

**Ans. **Let is true for every real number and

And There exists real numbers and for which .

Now There exists real numbers and for which therefore

**4. State whether the ‘or’ used in the following statements is “exclusive” or inclusive”. Give reasons for your answers.**

**(i) Sun rises or moon sets.**

**(ii) To apply for a driving license, you should have a ration card or a passport.**

**(iii) All integers are positive or negative.**

**Ans. (i) **This statement makes use of exclusive ‘or’. Since when sun rises, moon does not set during day-time.

**(ii) **This statement makes use of inclusive ‘or’. Since you can apply for a driving licence even if you have a ration card as well as passport.

**(iii) **This statement makes use of exclusive ‘or’. Since all integers are positive as well as negative.