91Ó°ÊÓ

Write the statements in 6-9 in symbolic form using the symbols \(\sim, \vee\), and \(\wedge\) and the indicated letters to represent component statements. Let \(h=\) "John is healthy," \(w=\) "John is wealthy," and \(s=\) "John is wise." a. John is healthy and wealthy but not wise. b. John is not wealthy but he is healthy and wise. c. John is neither healthy, wealthy, nor wise. d. John is neither wealthy nor wise, but he is healthy. e. John is wealthy, but he is not both healthy and wise.

Let \(p\) be the statement "DATAENDFLAG is off,"' \(q\) the statement "ERROR equals 0 ," and \(r\) the statement "SUM is less than 1,000." Express the following sentences in symbolic notation. a. DATAENDFLAG is off, ERROR equals 0 , and SUM is less than \(1,000 .\) b. DATAENDFLAG is off but ERROR is not equal to \(0 .\) c. DATAENDFLAG is off; however ERROR is not 0 or SUM is greater than or equal to 1,000 . d. DATAENDFLAG is on and ERROR equals 0 but SUM is greater than or equal to 1,000 . e. Either DATAENDFLAG is on or it is the case that both ERROR equals 0 and SUM is less than 1,000 .

In the following sentence, is the word or used in its inclusive or exclusive sense? A team wins the playoffs if it wins two games in a row or a total of three games.

Use a truth table to prove the validity of modus tollens. $$ \begin{aligned} & p \rightarrow q \\ & \sim q \\ \therefore & \sim p \end{aligned} $$

Use truth tables to show that the following forms of argument are invalid. a. \(\begin{array}{rlrl} & p \rightarrow q & \text { b. } & p & p \rightarrow q \\\ q & & & \sim p \\ & \therefore p & \therefore & \sim q \\ \text { (converse error) } & & \text { (inverse error) }\end{array}\)

Construct circuits for the Boolean expressions in 13-17. \(P \vee(\sim P \wedge \sim Q)\)

Construct circuits for the Boolean expressions in 13-17. \((P \wedge \sim Q) \vee(\sim P \wedge R)\)

Perform the arithmetic in \(13-20\) using binary notation. $$ \begin{array}{r} 10100_{2} \\ -\quad 1101_{2} \\ \hline \end{array} $$

Write truth tables for the statement forms in \(14-18\). $$ (p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r) $$

For each of the tables in 18-21, construct (a) a Boolean expression having the given table as its truth table and (b) a circuit having the given table as its input/output table. \begin{tabular}{|ccc|c|} \hline \(\boldsymbol{P}\) & \(\boldsymbol{Q}\) & \(\boldsymbol{R}\) & \(\boldsymbol{S}\) \\ \hline 1 & 1 & 1 & 0 \\ \hline 1 & 1 & 0 & 1 \\ \hline 1 & 0 & 1 & 0 \\ \hline 1 & 0 & 0 & 0 \\ \hline 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 0 & 0 \\ \hline 0 & 0 & 1 & 0 \\ \hline 0 & 0 & 0 & 0 \\ \hline \end{tabular}