. . . . 1.) . . . . Let proposition $ P $ be : It will rain. Let proposition $ Q $ be : It will hail. Then

It will rain or it will not hail.

equivalent to

$ P \vee ( \sim Q ) $

equivalent to (See Theorem 1.2 d.)

$ Q \Rightarrow P $

equivalent to

If it hails, then it will rain.

which is also equivalent to (See Theorem 1.1 a.)

If it does not rain, then it will not hail.


. . . . 2.) . . . . Let proposition $ P $ be : The food is cold. Let proposition $ Q $ be : The food is bad. Then

The food is cold or the food is bad.

equivalent to

$ P \vee Q $

equivalent to

$ P \vee ( \sim ( \sim Q )) $

equivalent to (See Theorem 1.2 d.)

$ ( \sim Q ) \Rightarrow P $

equivalent to

If the food is not bad, then the food is cold.

which is also equivalent to (See Theorem 1.1 a.)

If the food is not cold, then the food is bad.


. . . . 3.) . . . . Let proposition $ P $ be : She is tall. Let proposition $ Q $ be : She has brown eyes. Then

She is not tall or she does not have brown eyes.

equivalent to

$ ( \sim P ) \vee ( \sim Q ) $

equivalent to (See Theorem 1.2 d.)

$ ( \sim (\sim Q ) ) \Rightarrow ( \sim P ) $

equivalent to

$ Q \Rightarrow ( \sim P ) $

equivalent to

If she has brown eyes, then she is not tall.

which is also equivalent to (See Theorem 1.1 a.)

If she is tall, then she does not have brown eyes.


. . . . 4.) . . . . Let proposition $ P $ be : I will buy a new car. Let proposition $ Q $ be : I win the lottery. Then

I will buy a new car only if I win the lottery.

equivalent to

$ P $ only if $ Q $

equivalent to (See equivalent statements to $ P \Rightarrow Q $.)

$ P \Rightarrow Q $

equivalent to

If I buy a new car, then I won the lottery.

which is also equivalent to (See Theorem 1.1 a.)

If I don't win the lottery, then I won't buy a car.


. . . . 5.) . . . . Let proposition $ P $ be : He fails the biochemistry exam. Let proposition $ Q $ be : He studies all week. Then

He will fail the biochemistry exam unless he studies all week.

equivalent to

$ P $ unless $ Q $

equivalent to (See equivalent statements to $ \sim Q \Rightarrow P $.)

$ ( \sim Q ) \Rightarrow P $

equivalent to

If he does not study all week, then he will fail the biochemistry exam.

which is also equivalent to (See Theorem 1.1 a.)

If he passes the biochemistry exam, then he studied all week.


. . . . 6.) . . . . Let proposition $ P $ be : Tarzan will be very unhappy. Let proposition $ Q $ be : Jane is in his life. Then

Tarzan will be very unhappy without Jane in his life.

equivalent to

$ P $ without $ Q $

equivalent to (See equivalent statements to $ \sim Q \Rightarrow P $.)

$ ( \sim Q ) \Rightarrow P $

equivalent to

If Jane is not in his life, then Tarzan will be very unhappy.

which is also equivalent to (See Theorem 1.1 a.)

If Tarzan is happy, then Jane is in his life.


. . . . 7.) . . . . Let proposition $ P $ be : She won't go to the movie. Let proposition $ Q $ be : He goes with her. Then

She won't go to the movie unless he goes with her.

equivalent to

$ P $ unless $ Q $

equivalent to (See equivalent statements to $ \sim Q \Rightarrow P $.)

$ ( \sim Q ) \Rightarrow P $

equivalent to

If he does not go with her, then she will not go to the movie.

which is also equivalent to (See Theorem 1.1 a.)

If she goes to the movie, then he will go with her.


. . . . 8.) . . . . Let proposition $ P $ be : Milk is blue. Let proposition $ Q $ be : Bananas are red. Then

It is not true that milk is blue and bananas are red.

equivalent to

$ \sim (P \wedge Q) $

equivalent to (See Theorem 1.2 b.)

$ ( \sim P ) \vee ( \sim Q ) $

equivalent to (See Theorem 1.2 d.)

$ Q \Rightarrow ( \sim P ) $

equivalent to

If bananas are red, then milk is not blue.

which is also equivalent to (See Theorem 1.1 a.)

If milk is blue, then bananas are not red.


. . . . 9.) . . . . Let proposition $ P $ be : You will get an A. Let proposition $ Q $ be : You study. Then

You will get an A only if you study.

equivalent to

$ P $ only if $ Q $

equivalent to (See equivalent statements to $ P \Rightarrow Q $.)

$ P \Rightarrow Q $

equivalent to

If you get an A, then you studied.

which is also equivalent to (See Theorem 1.1 a.)

If you do not study, then you will not get an A.


. . . . 10.) . . . . Let proposition $ P $ be : You will get an A. Let proposition $ Q $ be : You study. Then

You will get an A whenever you study.

equivalent to

$ P $ whenever $ Q $

equivalent to (See equivalent statements to $ Q \Rightarrow P $.)

$ Q \Rightarrow P $

equivalent to

If you study, then you will get an A.

which is also equivalent to (See Theorem 1.1 a.)

If you do not get an A, then you did not study.


. . . . 11.) . . . . Let proposition $ P $ be : You will get an A. Let proposition $ Q $ be : You study. Then

You will get an A if you study.

equivalent to

$ P $ if $ Q $

equivalent to (See equivalent statements to $ Q \Rightarrow P $.)

$ Q \Rightarrow P $

equivalent to

If you study, then you will get an A.

which is also equivalent to (See Theorem 1.1 a.)

If you do not get an A, then you did not study.


. . . . 12.) . . . . Let proposition $ P $ be : You will get an A. Let proposition $ Q $ be : You study. Then

To get an A it is sufficient that you study.

equivalent to

$ P $ sufficient that $ Q $

equivalent to (See equivalent statements to $ P \Rightarrow Q $.)

$ P \Rightarrow Q $

equivalent to

If you get an A, then you studied.

which is also equivalent to (See Theorem 1.1 a.)

If you do not study, then you will not get an A.


. . . . 13.) . . . . Let proposition $ P $ be : You will get an A. Let proposition $ Q $ be : You study. Then

To get an A it is necessary that you study.

equivalent to

$ P $ necessary that $ Q $

equivalent to (See equivalent statements to $ Q \Rightarrow P $.)

$ Q \Rightarrow P $

equivalent to

If you study, then you will get an A.

which is also equivalent to (See Theorem 1.1 a.)

If you do not get an A, then you did not study.


RETURN to problem set.


Duane Kouba 2002-05-21