The Proof Page

by D. A. Kouba

Section 1.2- Conditionals and Biconditionals; Mathematically Equivalent Statements

In this section we will introduce two more mathematical connectives. This will allow for a wider and more useful range of propositional forms. In particular, we will now include in our propositional forms, sentences of the form ``If , then ", one of the most useful forms in the study of mathematics.

: Let and be propositions. The (read `` implies ." or ``If , then .") is true whenever is true or is false.

The following truth table illustrates all possible truth values for the conditional sentence .

: Let and be propositions. The (read `` if and only if .") is true exactly when and have the same truth values.

The following truth table illustrates all possible truth values for the biconditional sentence .

: Let and be propositions.

- 1.) The of is .
- 2.) The of is .

: Let and be propositions.

- a.) is equivalent to .
- b.) is NOT equivalent to .

- a.) Use a truth table .

- b.) Use a truth table .

- 1.) If a function is differentiable at , then the function is continuous at .
- 2.) If a function is not continuous at , then the function is not differentiable at .

- 1.) If I go to Mexico and lay on the beach, then I will get a suntan.

- 2.) If I have a suntan, then I went to Mexico and laid on the beach.

- a.)
is equivalent to
.

- b.)
is equivalent to
.

- c.)
is equivalent to
.

- d.)
is equivalent to
.

- e.)
is equivalent to
.

- f.)
is equivalent to
.

- g.)
is equivalent to
.

- h.) is equivalent to .

: Parts a.) and b.) in Theorem 1.2 make the connection between ``and" and ``or" statements. Part d.) makes the connection between ``if, then" and ``or" statements. Parts e.) and f.) represent associative properties for ``and" and ``or" statements. Parts g.) and h.) illustrate distributive properties.

Because the English language and use of common words can sometimes be ambiguous, we make the following mention of equivalent statements.

: On The Proof Page we will assume that the statement (If , then ) is equivalent to EACH of the following statements.

- 1.) if .

- 2.) only if .

- 3.) only when .

- 4.) whenever .

- 5.) when .

- 6.) implies .

- 7.) is sufficient for .

- 8.) is necessary for .

: The following sets of statements are equivalent.

- 1.) . . . . . . . a.) He will get wet if he stands in the rain.

. . . . . . . . . . . . . . b.) If he stands in the rain, then he will get wet.

- 2.) . . . . . . . a.) I will get a ticket for speeding only if I get caught.

. . . . . . . . . . . . . . b.) If I get a ticket for speeding, then I got caught.

- 3.) . . . . . . . a.) She takes a bike ride whenever she is upset.

. . . . . . . . . . . . . . b.) If she is upset, then she takes a bike ride.

- 4.) . . . . . . . a.) If you get eight hours of sleep, then you will feel good in the morning.

. . . . . . . . . . . . . . b.) Feeling good in the morning is necessary for getting eight hours of sleep.

. . . . . . . . . . . . . . . . . . c.) Getting eight hours of sleep is sufficient for feeling good in the morning.

: On The Proof Page we will assume that the statement (If not , then ) is equivalent to EACH of the following statements.

- 1.) unless .
- 2.) without .

: The following four statements are equivalent.

- 1.) . . . . She will not go swimming unless the water is very warm.

- 2.) . . . . She will not go swimming without the water being very warm.

- 3.) . . . . If the water is not very warm, then she will not go swimming.

- 4.) . . . . If she goes swimming, then the water is very warm. (See Theorem 1.1 a.)

: Rewrite each of the following sentences in symbolic propositional form. Then write each sentence in conditional (If ..., then ...) form. Find solutions HERE .

- 1.) It will rain or it will not hail.

- 2.) The food is cold or the food is bad.

- 3.) She is not tall or she does not have brown eyes.

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

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

- 6.) Tarzan will be very unhappy without Jane in his life.

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

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

- 9.) You will get an A only if you study.

- 10.) You will get an A whenever you study.

- 11.) You will get an A if you study.

- l2.) To get an A it is sufficient that you study.

- l3.) To get an A it is necessary that you study.

: Rewrite each of the following sentences in symbolic propositional form. Then use a truth table to prove that they are equivalent. Find solutions HERE .

- Sentence 1 : If wealth implies happiness, then you are materialistic.

- Sentence 2 : You are wealthy and not happy, or you are materialistic.

NOTE : (ORDER OF OPERATIONS) If parantheses are not used to clearly indicate the use of connectives, then invoke the connectives in the following order- , , , , . For example, is equivalent to .

: Provide parantheses to clarify the following ambiguous expressions. Find solutions HERE .

- 1.)

- 2.)

- 3.)

- 4.)

RETURN to The Proof Page .

Please e-mail your comments, questions, or suggestions to D. A. Kouba at
kouba@math.ucdavis.edu .