some k . Then n = k . Therefore, n is even. Example For the given statements identify the necessary and sufficient conditions.
t: If you drive over km per hour, then you will get a fine. Solution Let p and q denote the statements: p : you drive over km per hour. q : you will get a fine. The implication if p , then q indicates that p is sufficient for q .
That is driving over km per hour is sufficient to get a fine. Here the sufficient condition is “driving over km per hour”: Similarly, if p , then q also indicates that q is necessary for p . That is MATHEMATICAL REASONING When you drive over km per hour, you will necessarily get a fine. Here the necessary condition is “getting a fine”.
Miscellaneous Exercise on Chapter . Write the negation of the following statements: p: For every positive real number x , the number x – is also positive. q: All cats scratch. (iii) r: For every real number x , either x > or x < .
(iv) s: There exists a number x such that < x < . . State the converse and contrapositive of each of the following statements: p: A positive integer is prime only if it has no divisors other than and itself. q: I go to a beach whenever it is a sunny day.
(iii) r: If it is hot outside, then you feel thirsty. . Write each of the statements in the form “if p , then q ” p: It is necessary to have a password to log on to the server. q: There is traffic jam whenever it rains.
(iii) r: You can access the website only if you pay a subsciption fee. . Rewrite each of the following statements in the form “ p if and only if q ” p: If you watch television, then your mind is free and if your mind is free, then