91Ó°ÊÓ

Use 8-bit representations to compute the sums in 31-36. $$ 62+(-18) $$

a. Show that for the Sheffer stroke |, $$ P \wedge Q \equiv(P \mid Q) \mid(P \mid Q) . $$ b. Use the results of Example \(1.4 .7\) and part (a) above to write \(P \wedge(\sim Q \vee R)\) using only Sheffer strokes.

Show that the following logical equivalences hold for the c. \(P \wedge Q \equiv(P \downarrow P) \downarrow(Q \downarrow Q)\) Peirce arrow \(\downarrow\), where \(P \downarrow Q \equiv \sim(P \vee Q)\). \(H\) d. Write \(P \rightarrow Q\) using Peirce arrows only. a. \(\sim P \equiv P \downarrow P\) e. Write \(P \leftrightarrow Q\) using Peirce arrows only. b. \(P \vee Q \equiv(P \downarrow Q) \downarrow(P \downarrow Q)\)

Assume \(x\) is a particular real number and use De Morgan's laws to write negations for the statements in 35-38. $$ -2

Show that if \(a\) and \(b\) are integers in the range 1 through 128 , and the sum of \(a\) and \(b\) is also in this range, then \(2^{8} \leq\left(2^{8}-a\right)+\left(2^{8}-b\right)<2^{9}\). Explain why it follows that the binary representation of \(\left(2^{8}-a\right)+\left(2^{8}-b\right)\) has a leading 1 in the \(2^{8}\) th position.

In the back of an old cupboard you discover a note signed by a pirate famous for his bizarre sense of humor and love of logical puzzles. In the note he wrote that he had hidden treasure somewhere on the property. He listed five true statements (a-e below) and challenged the reader to use them to figure out the location of the treasure. a. If this house is next to a lake, then the treasure is not in the kitchen. b. If the tree in the front yard is an elm, then the treasure is in the kitchen. c. This house is next to a lake. d. The tree in the front yard is an elm or the treasure is buried under the flagpole. e. If the tree in the back yard is an oak, then the treasure is in the garage. Where is the treasure hidden?

Rewrite the statements in if-then form. This door will not open unless a security code is entered.

In 38-41 (a) use the logical equivalences \(p \rightarrow q \equiv \sim p \vee q\) and \(p \leftrightarrow q \equiv(\sim p \vee q) \wedge(\sim q \vee p)\) to rewrite the given statement forms without using the symbol \(\rightarrow\) or \(\leftrightarrow\), and (b) use the logical equivalence \(p \vee q \equiv \sim(\sim p \wedge \sim q)\) to rewrite each statement form using only \(\wedge\) and \(\sim\). $$ p \wedge \sim q \rightarrow r $$

Sharky, a leader of the underworld, was killed by one of his own band of four henchmen. Detective Sharp interviewed the men and determined that all were lying except for one. He deduced who killed Sharky on the basis of the following statements: a. Socko: Lefty killed Sharky. b. Fats: Muscles didn't kill Sharky. c. Lefty: Muscles was shooting craps with Socko when Sharky was knocked off. d. Muscles: Lefty didn't kill Sharky. Who did kill Sharky?

In 38-41 (a) use the logical equivalences \(p \rightarrow q \equiv \sim p \vee q\) and \(p \leftrightarrow q \equiv(\sim p \vee q) \wedge(\sim q \vee p)\) to rewrite the given statement forms without using the symbol \(\rightarrow\) or \(\leftrightarrow\), and (b) use the logical equivalence \(p \vee q \equiv \sim(\sim p \wedge \sim q)\) to rewrite each statement form using only \(\wedge\) and \(\sim\). $$ (p \rightarrow(q \rightarrow r)) \leftrightarrow((p \wedge q) \rightarrow r) $$