\displaystyle \textbf{Question 1: }\text{Write each of the following statements in the form ``if }p,\text{ then }q\text{''.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) You can access the website only if you pay a subscription fee.}
\displaystyle \text{If you pay a subscription fee, then you can access the website.}

\displaystyle \text{(ii) There is a traffic jam whenever it rains.}
\displaystyle \text{If it rains, then there is a traffic jam.}

\displaystyle \text{(iii) It is necessary to have a passport to log on to the server.}
\displaystyle \text{If you log on to the server, then you have a passport.}

\displaystyle \text{(iv) It is necessary to be rich in order to be happy.}
\displaystyle \text{If you are happy, then you are rich.}

\displaystyle \text{(v) The game is cancelled only if it is raining.}
\displaystyle \text{If the game is cancelled, then it is raining.}

\displaystyle \text{(vi) It rains only if it is cold.}
\displaystyle \text{If it rains, then it is cold.}

\displaystyle \text{(vii) Whenever it rains, it is cold.}
\displaystyle \text{If it rains, then it is cold.}

\displaystyle \text{(viii) It never rains when it is cold.}
\displaystyle \text{If it is cold, then it does not rain.}
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{State the converse and contrapositive of each of the following statements.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) If it is hot outside, then you feel thirsty.}
\displaystyle \text{Converse: If you feel thirsty, then it is hot outside.}
\displaystyle \text{Contrapositive: If you do not feel thirsty, then it is not hot outside.}

\displaystyle \text{(ii) I go to the beach whenever it is a sunny day.}
\displaystyle \text{Converse: If I go to the beach, then it is a sunny day.}
\displaystyle \text{Contrapositive: If I do not go to the beach, then it is not a sunny day.}

\displaystyle \text{(iii) A positive integer is prime only if it has no divisors other than }1\text{ and itself.}
\displaystyle \text{Converse: If a positive integer has no divisors other than }1\text{ and itself,}
\displaystyle \text{then it is a prime number.}
\displaystyle \text{Contrapositive: If a positive integer has divisors other than }1\text{ and itself,}
\displaystyle \text{then it is not a prime number.}

\displaystyle \text{(iv) If you live in Delhi, then you have winter clothes.}
\displaystyle \text{Converse: If you have winter clothes, then you live in Delhi.}
\displaystyle \text{Contrapositive: If you do not have winter clothes, then you do not live in Delhi.}

\displaystyle \text{(v) If a quadrilateral is a parallelogram, then its diagonals bisect each other.}
\displaystyle \text{Converse: If the diagonals of a quadrilateral bisect each other,}
\displaystyle \text{then it is a parallelogram.}
\displaystyle \text{Contrapositive: If the diagonals of a quadrilateral do not bisect each other,}
\displaystyle \text{then it is not a parallelogram.}
\displaystyle \\

\displaystyle \textbf{Question 3: }\text{Rewrite each of the following statements in the form}
\displaystyle \text{``}p\text{ if and only if }q\text{''.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) If you watch television, then your mind is free, and if your mind is free,}
\displaystyle \text{then you watch television.}
\displaystyle \therefore \text{You watch television if and only if your mind is free.}

\displaystyle \text{(ii) If a quadrilateral is equiangular, then it is a rectangle, and if a quadrilateral}
\displaystyle \text{is a rectangle, then it is equiangular.}
\displaystyle \therefore \text{A quadrilateral is a rectangle if and only if it is equiangular.}

\displaystyle \text{(iii) To get an A grade, it is necessary and sufficient that you do all your}
\displaystyle \text{homework regularly.}
\displaystyle \therefore \text{You get an A grade if and only if you do all your homework regularly.}

\displaystyle \text{(iv) If a tumbler is half empty, then it is half full, and if a tumbler is half full,}
\displaystyle \text{then it is half empty.}
\displaystyle \therefore \text{A tumbler is half empty if and only if it is half full.}
\displaystyle \\

\displaystyle \textbf{Question 4: }\text{Determine the contrapositive of each of the following statements.}
\displaystyle \text{Answer:}

\displaystyle \text{(i) If Mohan is a poet, then he is poor.}
\displaystyle \text{Contrapositive: If Mohan is not poor, then he is not a poet.}

\displaystyle \text{(ii) Only if Max studies will he pass the test.}
\displaystyle \text{Equivalently, if Max passes the test, then he studies.}
\displaystyle \text{Contrapositive: If Max does not study, then he will not pass the test.}

\displaystyle \text{(iii) If she works, then she will earn money.}
\displaystyle \text{Contrapositive: If she does not earn money, then she does not work.}

\displaystyle \text{(iv) If it snows, then they do not drive the car.}
\displaystyle \text{Contrapositive: If they drive the car, then it does not snow.}

\displaystyle \text{(v) It never rains when it is cold.}
\displaystyle \text{Equivalently, if it is cold, then it does not rain.}
\displaystyle \text{Contrapositive: If it rains, then it is not cold.}

\displaystyle \text{(vi) If Ravish skis, then it snowed.}
\displaystyle \text{Contrapositive: If it did not snow, then Ravish does not ski.}

\displaystyle \text{(vii) If }x<0,\text{ then }x\text{ is not positive.}
\displaystyle \text{Contrapositive: If }x\text{ is positive, then }x\text{ is not less than zero.}

\displaystyle \text{(viii) If he has courage, then he will win.}
\displaystyle \text{Contrapositive: If he does not win, then he does not have courage.}

\displaystyle \text{(ix) It is necessary to be strong in order to be a sailor.}
\displaystyle \text{Equivalently, if a person is a sailor, then the person is strong.}
\displaystyle \text{Contrapositive: If a person is not strong, then the person cannot be a sailor.}

\displaystyle \text{(x) Only if he does not tire will he win.}
\displaystyle \text{Equivalently, if he wins, then he does not tire.}
\displaystyle \text{Contrapositive: If he tires, then he will not win.}

\displaystyle \text{(xi) For an integer }x,\text{ if }x^2\text{ is odd, then }x\text{ is odd.}
\displaystyle \text{Contrapositive: If }x\text{ is even, then }x^2\text{ is even.}
\displaystyle \\

 

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