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**Miscellaneous Exercise**

**1. Write the negation of the following statements:**

**(i) **** For every positive real number ****, the number **** is also positive.**

**(ii) **** All cats scratch.**

**(iii) **** For every real number ****, either **** or **

**(iv) **** There exists a number **** such that **

**Ans. (i) ** There exists a positive real number such that is not positive.

**(ii) ** There exists a cat which does not scratch.

**(iii) ** There exists a real number such that or

**(iv) ** There does not exist a number such that

**2. State the converse and contrapositive of each of the following statements:**

**(i) **** A positive integer is prime only if it has no divisors other than 1 and itself.**

**(ii) **** I go to a beach whenever it is a sunny day.**

**(iii) **** If it is not outside, then you feel thirsty.**

**Ans. (i) **Contrapositive: If a positive integer has divisors other than 1 and itself then it is not prime.

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

**(ii) **Contrapositive: If I do not go to a beach then it is not a sunny day.

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

**(iii) **Contrapositive: If you do not feel thirsty then it is not hot outside.

Converse: If you feel thirsty then it is hot outside.

**3. Rewrite each of the following statements in the form “**** it and only if ****”.**

**(i) It is necessary to have a password to log on to the server.**

**(ii) There is traffic Jam whenever it rains.**

**(iii) You can access the website only if you pay a subscription fee.**

**Ans. (i) **If you log on to the server then you have a password.

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

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

**4. Rewrite each of the following statements in the form “**** it and only if ****”.**

**(i) **** If you watch television, then your mind is free and if your mind is free, then you watch television.**

**(ii) **** For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.**

**(iii) **** If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.**

**Ans. (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 and only if it is a rectangle.

**5. Given below are two statements:**

** 25 is a multiple of 5.**

** 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.**

**Ans. **The compound statement with “and” is: 25 is a multiple of 5 and 8.

Since, is true and is false therefore the compound statement with “and” is not true.

Therefore, the statement “ and ” is not valid.

Now the compound statement with “or” is: 25 is a multiple of 5 or 8.

Since, is true and is false therefore the compound statement with “or” is true.

Therefore, the statement “ and ” is valid.

**6. Check the validity of the statements given below by the method given against it:**

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

**(ii) **** If **** is a real number with **** then **** (by contradiction method)**

**Ans. (i) **Let us assume that is not true.

Sum of an irrational and a rational number is not irrational.

There exists an irrational number and a rational number such that is not irrational.

(say) is a rational number.

is rational.

But is irrational, which is contradiction, therefore our supposition is wrong.

Therefore, is true.

**(ii) **Let and be the statements given by

is a real number with

If possible let is not true, then is true.

and is true.

is a real number with and which is contradiction, therefore our supposition is wrong.

Therefore, is true.

**7. Write the following statement in five different ways, conveying the same meaning:**

** If a triangle is equiangular, then it is an obtuse angled triangle.**

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

**(ii) **A triangle is equiangular only if it an obtuse angled triangle.

**(iii) **For a triangle to be equiangular it is necessary it is an obtuse angled triangle.

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

**(v) **Is a triangle is not equiangular then it is not an obtuse angled triangle.