91Ó°ÊÓ

Given any statement form, is it possible to find a logically equivalent form that uses only \(\sim\) and \(\wedge ?\) Justify your answer.

In \(41-44\) a set of premises and a conclusion are given. Use the valid argument forms listed in Table \(1.3 .1\) to deduce the conclusion from the premises, giving a reason for each step as in Example 1.3.10. Assume all variables are statement variables. a. \(p \vee q\) b. \(\quad q \rightarrow r\) c. \(\quad p \wedge s \rightarrow t\) d. \(\quad \sim r\) e. \(\sim q \rightarrow u \wedge s\) f. \(\therefore t\)

Convert the integers in \(44-46\) from binary to hexadecimal notation. $$ 00101110_{2} $$

Use truth tables to establish which of the statement forms in 41-44 are tautologies and which are contradictions. $$ (\sim p \vee q) \vee(p \wedge \sim q) $$

A sufficient condition for Jon's team to win the championship is that it win the rest of its games.

In addition to binary and hexadecimal, computer scientists also use octal notation (base 8) to represent numbers. Octal notation is based on the fact that any integer can be uniquely represented as a sum of numbers of the form \(d \cdot 8^{n}\), where each \(n\) is a nonnegative integer and each \(d\) is one of the integers from 0 to 7 . Thus, for example, \(5073_{8}=5 \cdot 8^{3}+0 \cdot 8^{2}+7 \cdot 8^{1}+3 \cdot 8^{0}=2619_{10} .\) a. Convert \(61502_{8}\) to decimal notation. b. Convert \(20763_{8}\) to decimal notation. c. Describe methods for converting integers from octal to binary notation and the reverse that are similar to the methods used in Examples 1.5.12 and 1.5.13 for converting back and forth from hexadecimal to binary notation. Give examples showing that these methods result in correct answers.

"If compound \(X\) is boiling, then its temperature must be a least \(150^{\circ} \mathrm{C}\)." Assuming that this statement is true, which of the following must also be true? a. If the temperature of compound \(X\) is at least \(150^{\circ} \mathrm{C}\), then compound \(X\) is boiling. b. If the temperature of compound \(X\) is less than \(150^{\circ} \mathrm{C}\), then compound \(X\) is not boiling. c. Compound \(X\) will boil only if its temperature is at least \(150^{\circ} \mathrm{C}\). d. If compound \(X\) is not boiling, then its temperature is less than \(150^{\circ} \mathrm{C}\). e. A necessary condition for compound \(X\) to boil is that its temperature be at least \(150^{\circ} \mathrm{C}\). f. A sufficient condition for compound \(X\) to boil is that its temperature be at least \(150^{\circ} \mathrm{C}\).

In Example 1.1.5, the symbol \(\oplus\) was introduced to denote exclusive or, so \(p \oplus q \equiv(p \vee q) \wedge \sim(p \wedge q)\). Hence the truth table for exclusive or is as follows: $$ \begin{array}{|cc|c|} \hline p & q & p \oplus q \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{T} \\ \hline \mathrm{F} & \mathrm{F} & \mathrm{F} \\ \hline \end{array} $$ a. Find simpler statement forms that are logically equivalent to \(p \oplus p\) and \((p \oplus p) \oplus p\). b. Is \((p \oplus q) \oplus r \equiv p \oplus(q \oplus r) ?\) Justify your answer. c. Is \((p \oplus q) \wedge r \equiv(p \wedge r) \oplus(q \wedge r) ?\) Justify your answer.

In logic and in standard English, a double negative is equivalent to a positive. Is there any English usage in which a double positive is equivalent to a negative? Explain.