Section1pt2

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.

$\underline { \rm DEFINITION }$ : Let

and

be propositions. The $\underline { \rm conditional \ sentence }$ $P \Rightarrow Q$ (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 $P \Rightarrow Q$ .

		$P \Rightarrow Q$

$\underline { \rm DEFINITION }$ : Let

and

be propositions. The $\underline { \rm biconditional \ sentence }$ $P \Leftrightarrow Q$ (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 $P \Leftrightarrow Q$ .

		$P \Leftrightarrow Q$

$\underline { \rm DEFINITION }$ : Let

and

be propositions.

1.) The $\underline { \rm converse }$ of $P \Rightarrow Q$ is $Q \Rightarrow P$ .
2.) The $\underline { \rm contrapositive }$ of $P \Rightarrow Q$ is $( \sim Q) \Rightarrow ( \sim P)$ .

The following two theorems present a list of equivalent propositions. These equivalences will allow us to easily change from one propositional form to another.

$\underline { \rm THEOREM \ 1.1 }$ : Let

and

be propositions.

a.) $P \Rightarrow Q$ is equivalent to $( \sim Q) \Rightarrow ( \sim P)$ .
b.) $P \Rightarrow Q$ is NOT equivalent to $Q \Rightarrow P$ .

$\underline { \rm PROOF }$ :

a.) Use a truth table .
b.) Use a truth table .

$\underline { \rm EXAMPLE }$ : Equivalence of the following statements follows from Theorem 1.1 a.).

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 .

$\underline { \rm EXAMPLE }$ : The following statements are not equivalent. See Theorem 1.1 b.).

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.

$\underline { \rm THEOREM \ 1.2 }$ : Let

and

be propositions.

a.) $P \Leftrightarrow Q$ is equivalent to $(P \Rightarrow Q) \wedge (Q \Rightarrow P)$ .
b.) $\sim (P \wedge Q)$ is equivalent to $(\sim P) \vee (\sim Q)$ .
c.) $\sim (P \vee Q)$ is equivalent to $(\sim P) \wedge (\sim Q)$ .
d.) $P \Rightarrow Q$ is equivalent to $Q \vee (\sim P)$ .
e.) $P \vee ( Q \vee R)$ is equivalent to $( P \vee Q ) \vee R$ .
f.) $P \wedge ( Q \wedge R)$ is equivalent to $( P \wedge Q ) \wedge R$ .
g.) $P \wedge ( Q \vee R)$ is equivalent to $( P \wedge Q ) \vee ( P \wedge R )$ .
h.) $P \vee ( Q \wedge R)$ is equivalent to $( P \vee Q ) \wedge ( P \vee R )$ .

$\underline { \rm PROOF }$ : All are easily proven using truth tables.

$\underline { \rm NOTE }$ : 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.

$\underline { \rm NOTE }$ : On The Proof Page we will assume that the statement $P \Rightarrow Q$ (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 .

$\underline { \rm EXAMPLE }$ : 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.

$\underline { \rm NOTE }$ : On The Proof Page we will assume that the statement $\sim Q \Rightarrow P$ (If not

, then

) is equivalent to EACH of the following statements.

1.) unless .
2.) without .

$\underline { \rm EXAMPLE }$ : 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.)

$\underline { \rm EXAMPLE }$ : 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.

$\underline { \rm EXAMPLE }$ : 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- $\sim$ , $\wedge$ , $\vee$ , $\Rightarrow$ , $\Leftrightarrow$ . For example, $P \Rightarrow Q \wedge R \Leftrightarrow S$ is equivalent to $(P \Rightarrow (Q \wedge R)) \Leftrightarrow S$ .

$\underline { \rm EXAMPLE }$ : Provide parantheses to clarify the following ambiguous expressions. Find solutions HERE .

1.) $\sim P \Leftrightarrow Q \Rightarrow R$
2.) $P \wedge \sim Q \Rightarrow R \vee S \Leftrightarrow \sim T$
3.) $P \Leftrightarrow Q \Rightarrow R \Rightarrow S \Leftrightarrow T$
4.) $P \vee Q \Rightarrow \sim R \Leftrightarrow S \wedge T$

RETURN to The Proof Page .

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

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Duane Kouba 2002-05-11