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Question-1 :-  Write the negation of the following statements:
(i) p: For every positive real number x, the number x – 1 is also positive.
(ii) q: All cats scratch.
(iii) r: For every real number x, either x > 1 or x < 1.
(iv) s: There exists a number x such that 0 < x < 1.

Solution :-
(i) The negation of statement p is as follows.
  There exists a positive real number x, such that x – 1 is not positive. 
        
(ii) The negation of statement q is as follows.
  There exists a cat that does not scratch.

(iii) The negation of statement r is as follows.
  There exists a real number x, such that neither x > 1 nor x < 1. 
        
(iv) The negation of statement s is as follows.
  There does not exist a number x, such that 0 < x < 1.
   

Question-2 :-  State the converse and contrapositive of each of the following statements:
(i) p: A positive integer is prime only if it has no divisors other than 1 and itself.
(ii) q: I go to a beach whenever it is a sunny day.
(iii) r: If it is hot outside, then you feel thirsty.

Solution :-
(i) Statement p can be written as follows.
  If a positive integer is prime, then it has no divisors other than 1 and itself. 
  The converse of the statement is as follows.
  If a positive integer has no divisors other than 1 and itself, then it is prime. 
  The contrapositive of the statement is as follows.
  If 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 as follows.
  If I go to a beach, then it is a sunny day.
  The contrapositive of the statement is as follows.
  If I do not go to a beach, then it is not a sunny day.
    
(iii) The converse of statement r is as follows.
  If you feel thirsty, then it is hot outside.
  The contrapositive of statement r is as follows.
  If you do not feel thirsty, then it is not hot outside.
   

Question-3 :-  Write each of the statements in the form “if p, then q”
(i) p: It is necessary to have a password to log on to the server.
(ii) q: There is traffic jam whenever it rains.
(iii) r: You can access the website only if you pay a subsciption fee.

Solution :-
(i) Statement p can be written as follows.
  If you log on to the server, then you have a password. 
        
(ii) Statement q can be written as follows.
  If it rains, then there is a traffic jam.

(iii) Statement r can be written as follows.
  If you can access the website, then you pay a subscription fee.
    

Question-4 :-  Rewrite each of the following statements in the form “p if and only if q”
(i) p: If you watch television, then your mind is free and if your mind is free, then you watch television.
(ii) q: For you to get an A grade, it is necessary and sufficient that you do all the homework regularly.
(iii) r: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.

Solution :-
(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.
   

Question-5 :- Given below are two statements
p : 25 is a multiple of 5.
q : 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.

Solution :-
  The compound statement with ‘And’ is “25 is a multiple of 5 and 8”.
  This is a false statement, since 25 is not a multiple of 8.
  The compound statement with ‘Or’ is “25 is a multiple of 5 or 8”.
  This is a true statement, since 25 is not a multiple of 8 but it is a multiple of 5.
   

Question-6 :-  Check the validity of the statements given below by the method given against it.
(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).
(ii) q: If n is a real number with n > 3, then n2 > 9 (by contradiction method).

Solution :-
(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 given statement, p, is false. 
  That is, we assume that the sum of an irrational number and a rational number is rational.
  Therefore, √a + b/c = d/e, where √a is irrational and b, c, d, e are integers.
  d/e - b/c is a rational number and √a	is an irrational number. 
  This is a contradiction. Therefore, our assumption is wrong.
  Therefore, the sum of an irrational number and a rational number is rational. 
  Thus, 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.
  That is, n² < 9
  Then, n > 3 and n is a real number.
  Squaring both the sides, we obtain n² > (3)²
  n² > 9, which is a contradiction, since we have assumed that n² < 9.
  Thus, the given statement is true. 
  That is, if n is a real number with n > 3, then n² > 9.
   

Question-7 :-  Write the following statement in five different ways, conveying the same meaning.
p: If a triangle is equiangular, then it is an obtuse angled triangle.

Solution :-
  The given statement can be written in five different ways as follows. 
(i) A triangle is equiangular implies that it is an obtuse-angled triangle. 
(ii) A triangle is equiangular only if it 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.
   
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