Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words explaining how the truth table supports your answer.
If Sam bought it at Crown Books, then Sam didn't pay full price.
Sam bought it at Crown Books or Sam paid full price.

Question

Write the following two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words explaining how the truth table supports your answer. 
If Sam bought it at Crown Books, then Sam didn't pay full price.
Sam bought it at Crown Books or Sam paid full price.

Quizplus · Accepted Answer

The Answer is Solution 1: The given statements are not logically equivalent. Let p be "Sam bought it at Crown Books," and q be "Sam didn't pay full price." Then the two statements have the following form:

p ightarrow q \quad ext { and } \quad p \vee \sim q

The truth tables for these statement forms are

\begin{array} { | l | l | l | | l | | l | } \hline p & q & \sim q & p ightarrow q & p \vee \sim q \\hline \hline T & T & F & T & T \T & F & T & F & T \F & T & F & T & F \F & F & T & T & T \\hline\end{array}

Rows 2 and 3 of the table show that

p ightarrow q

and

p \vee \sim q

do not always have the same truth values, and so

p ightarrow q ot eq p \vee \sim q

.
Solution 2: The given statements are not logically equivalent. Let p be "Sam bought it at Crown
Books," and q be "Sam paid full price." Then the two statements have the following form:

p ightarrow \sim q \quad ext { and } \quad p \vee q

The truth tables for these statement forms are

\begin{array} { | l | l | l | | l | | l | } \hline p & q & \sim q & p ightarrow \sim q & p \vee q \\hline \hline T & T & F & F & T \T & F & T & T & T \F & T & F & T & T \F & F & T & T & F \\hline\end{array}

Rows 1 and 4 of the table show that

p ightarrow \sim q

and

p \vee q

do not always have the same truth values, and so

p ightarrow \sim q ot \equiv p \vee q

.

Write the Following Two Statements in Symbolic Form and Determine