91Ó°ÊÓ

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

Write negations for each of the following statements. (Assume that all variables represent fixed quantities or entities, as appropriate.) a. If \(P\) is a square, then \(P\) is a rectangle. b. If today is New Year's Eve, then tomorrow is January. c. If the decimal expansion of \(r\) is terminating, then \(r\) is rational. d. If \(n\) is prime, then \(n\) is odd or \(n\) is 2 . e. If \(x\) is nonnegative, then \(x\) is positive or \(x\) is 0 . f. If Tom is Ann's father, then Jim is her uncle and Sue is her aunt. g. If \(n\) is divisible by 6 , then \(n\) is divisible by 2 and \(n\) is divisible by \(3 .\)

Suppose that \(p\) and \(q\) are statements so that \(p \rightarrow q\) is false. Find the truth values of each of the following: a. \(\sim p \rightarrow q\) b. \(p \vee q\) c. \(q \rightarrow p\)

Design a circuit to take input signals \(P, Q\), and \(R\) and output a 1 if, and only if, \(P\) and \(Q\) have the same value and \(Q\) and \(R\) have opposite values.

Design a circuit to take input signals \(P, Q\), and \(R\) and output a 1 if, and only if, all three of \(P, Q\), and \(R\) have the same value.

Determine which of the pairs of statement forms in \(19-28\) are logically equivalent. Justify your answers using truth tables and include a few words of explanation. Read \(\mathbf{t}\) to be a tautology and \(\mathbf{c}\) to be a contradiction. $$ (p \wedge q) \wedge r \text { and } p \wedge(q \wedge r) $$

A conditional statement is not logically equivalent to its converse.

Some of the arguments in 24-32 are valid, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. If Jules solved this problem correctly, then Jules obtained the answer 2 . Jules obtained the answer 2 . Jules solved this problem correctly.

The lights in a classroom are controlled by two switches: one at the back and one at the front of the room. Moving either switch to the opposite position turns the lights off if they are on and on if they are off. Assume the lights have been installed so that when both switches are in the down position, the lights are off. Design a circuit to control the switches.

A conditional statement is not logically equivalent to its inverse.