- Which of the following sentences are statements?a. The moon is made of green cheese.b. He is certainly a tall man.c. Two is a prime number.d. The game will be over by 4:00.e. Next year interest rates
- What is the truth value of each of the following statements?a. 8 is even or 6 is odd.b. 8 is even and 6 is odd.c. 8 is odd or 6 is odd.d. 8 is odd and 6 is odd.e. If 8 is odd, then 6 is odd.f. If 8
- Given the truth values \(A\) true, \(B\) false, and \(C\) true, what is the truth value of each of the following wffs?a. \(A \wedge(B \vee C)\)b. \((A \wedge B) \vee C\)c. \((A \wedge B)^{\prime}
- Given the truth values \(A\) false, \(B\) true, and \(C\) true, what is the truth value of each of the following wffs?a. \(A \rightarrow(B \vee C)\)b. \((A \vee B) \rightarrow C\)c. \(C
- Rewrite each of the following statements in the form "If \(A\), then \(B\)."a. Healthy plant growth follows from sufficient water.b. Increased availability of information is a necessary condition for
- Rewrite each of the following statements in the form "If \(A\), then \(B\)."a. Candidate Lu winning the election will be a sufficient condition for property taxes to increase.b. The user clicks Pause
- Common English has many ways to describe logical connectives. Write a wff for each of the following expressions.a. Either \(A\) or \(B\)b. Neither \(A\) nor \(B\)
- Common English has many ways to describe logical connectives. Write a wff for each of the following expressions.a. \(B\) whenever \(A\)b. \(A\) is derived from \(B\)c. \(A\) indicates \(B\)d. \(A\)
- Several forms of negation are given for each of the following statements. Which are correct?a. The answer is either 2 or 3.1. Neither 2 nor 3 is the answer.2. The answer is not 2 or not 3.3. The
- Several forms of negation are given for each of the following statements. Which are correct?a. The carton is sealed or the milk is sour.1. The milk is not sour or the carton is not sealed.2. The
- Write the negation of each statement.a. If the food is good, then the service is excellent.b. Either the food is good or the service is excellent.c. Either the food is good and the service is
- Write the negation of each statement.a. The processor is fast but the printer is slow.b. The processor is fast or else the printer is slow.c. If the processor is fast, then the printer is slow.d.
- Using the letters indicated for the component statements, translate the following compound statements into symbolic notation.a. A: prices go up; B: housing will be plentiful; C: housing will be
- Using the letters indicated for the component statements, translate the following compound statements into symbolic notation.a. A: the tractor wins; B: the truck wins; C: the race will be
- Let A, B, and C be the following statements:A Roses are red.B Violets are blue.C Sugar is sweet.Translate the following compound statements into symbolic notation.a. Roses are red and violets are
- Let A, B, and C, and D be the following statements:A The villain is French.B The hero is American.C The heroine is British.D The movie is good.Translate the following compound statements into
- Use \(A, B\), and \(C\) as defined in Exercise 15 to translate the following statements into English.Data from in exercise 15Let A, B, and C be the following statements:A Roses are red.B Violets are
- Use \(A, B\), and \(C\) as defined in Exercise 16 to translate the following statements into English.Data from in Exercise 16Let A, B, and C, and D be the following statements:A The villain is
- Using letters \(H, K, A\) for the component statements, translate the following compound statements into symbolic notation.a. If the horse is fresh, then the knight will win.b. The knight will win
- Using letters A, T, E for the component statements, translate the following compound statements into symbolic notation.a. If Anita wins the election, then tax rates will be reduced.b. Tax rates will
- Using letters F, B, S for the component statements, translate the following compound statements into symbolic notation.a. Plentiful fish are a sufficient condition for bears to be happy.b. Bears are
- Using letters P, C, B, L for the component statements, translate the following compound statements into symbolic notation.a. If the project is finished soon, then the client will be happy and the
- Construct truth tables for the following wffs. Note any tautologies or contradictions.a. \((A \rightarrow B) \leftrightarrow A^{\prime} \vee B\)b. \((A \wedge B) \vee C \rightarrow A \wedge(B \vee
- Construct truth tables for the following wffs. Note any tautologies or contradictions.a. \(A \rightarrow(B \rightarrow A)\)b. \(A \wedge B \leftrightarrow B^{\prime} \vee A^{\prime}\)c. \(\left(A
- Verify the equivalences in the list on page 9 by constructing truth tables. (We have already verified \(1 \mathrm{a}, 4 \mathrm{~b}\), and \(5 \mathrm{a}\).)
- Verify by constructing truth tables that the following wffs are tautologies. Note that the tautologies in parts \(\mathrm{b}, \mathrm{e}, \mathrm{f}\), and \(\mathrm{g}\) produce equivalences such as
- Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that will convert the left side into the right side. You may use any of the equivalencies in the
- Prove the following tautologies by starting with the left side and finding a series of equivalent wffs that will convert the left side into the right side. You may use any of the equivalencies in the
- We mentioned that \((A \wedge B) \vee C\) cannot be proved equivalent to \(A \wedge(B \vee C)\) using either of the associative tautological equivalences, but perhaps it can be proved some other way.
- Let \(P\) be the wff \(A \rightarrow B\). Prove or disprove whether \(P\) is equivalent to any of the following related wffs.a. the converse of \(P, B \rightarrow A\)b. the inverse of \(P, A^{\prime}
- Write a logical expression for a Web search engine to find sites pertaining to dogs that are not retrievers.
- Write a logical expression for a Web search engine to find sites pertaining to oil paintings by Van Gogh or Rembrandt but not Vermeer.
- Write a logical expression for a Web search engine to find sites pertaining to novels or plays about AIDS.
- Write a logical expression for a Web search engine to find sites pertaining to coastal wetlands in Louisiana but not in Alabama.
- Consider the following pseudocode.The input values for x are 1.0, 5.1, 2.4, 7.2, and 5.3. What are the output values? repeat i = 1 read a value for x if ((x < 5.0) and (2x < 10.7)) or (5x>5.1) then
- Suppose that A, B, and C represent conditions that will be true or false when a certain computer program is executed. Suppose further that you want the program to carry out a certain task only when A
- Rewrite the following statement form with a simplified conditional expression, where the function returns true if is odd. if not((Value1 or (not(Value1 statement 1 else end if statement2 < Value2)
- You want your program to execute statement 1 when AA is false, BB is false, and CC is true, and to execute statement 2 otherwise. You wrote \section*{if not(Anot(A and B)B) and CC then}statement
- Verify that \(A \rightarrow B\) is equivalent to \(A^{\prime} \vee B\).
- a. Using Exercise 39 and other equivalences, prove that the negation of \(A \rightarrow B\) is equivalent to \(A \wedge B^{\prime}\)b. Write the negation of the statement "If Sam passed his bar exam,
- Use algorithm TautologyTest to prove that the following expressions are tautologies.a. \(\left[B^{\prime} \wedge(A \rightarrow B)\right] \rightarrow A^{\prime}\)b. \([(A \rightarrow B) \wedge A]
- Use algorithm TautologyTest to prove that the following expressions are tautologies.a. \((A \wedge B) \wedge B^{\prime} \rightarrow A\)b. \(\left(A \wedge B^{\prime}\right) \rightarrow(A \rightarrow
- A memory chip from a digital camera has \(2^{5}\) bistable (ON-OFF) memory elements. What is the total number of ON-OFF configurations?
- In each case, construct compound wffs \(P\) and \(Q\) so that the given statement is a tautology.a. \(P \wedge Q\)b. \(P \rightarrow P^{\prime}\)c. \(P \wedge\left(Q \rightarrow P^{\prime}\right)\)
- From the truth table for \(A \vee B\), the value of \(A \vee B\) is true if \(A\) is true, if \(B\) is true, or if both are true. This use of the word "or," where the result is true if both
- Prove that \(A \oplus B \leftrightarrow(A \leftrightarrow B)^{\prime}\) is a tautology. Explain why this makes sense.
- Every compound statement is equivalent to a statement using only the connectives of conjunction and negation. To see this, we need to find equivalent wffs for \(A \vee B\) and for \(A \rightarrow B\)
- Show that every compound wff is equivalent to a wff using only the connectives of \(\vee\) and '.Exercises 47-50 show that defining four basic logical connectives (conjunction, disjunction,
- Show that every compound wff is equivalent to a wff using only the connectives of \(\rightarrow\) and '.Exercises 47-50 show that defining four basic logical connectives (conjunction, disjunction,
- Prove that there are compound statements that are not equivalent to any statement using only the connectives \(\rightarrow\) and \(\vee\).Exercises 47-50 show that defining four basic logical
- The binary connective \(\mid\) is called the Sheffer stroke, named for the American logic professor Henry Sheffer, who proved in 1913 that this single connective is the only one needed. The truth
- The binary connective \(\downarrow\) is called the Peirce arrow, named for American philosopher Charles Peirce (not for the antique automobile). The truth table for \(\downarrow\) is given here. In
- Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of two values, False or True. Three-valued logic allows a third value of Null or "unknown"
- Propositional wffs and truth tables belong to a system of two-valued logic because everything has one of two values, \(\mathrm{F}\) or \(\mathrm{T}\), which we can think of as 0 or 1 . In fuzzy
- In a three-valued logic system as described in Exercise 53, how many rows are needed for a truth table with \(n\) statement letters?
- In 2003, then U.S. Secretary of Defense Donald Rumsfeld won Britain's Plain English Campaign 2003 Golden Bull Award for this statement: "Reports that say that something hasn't happened are always
- Four machines, \(A, B, C\), and \(D\), are connected on a computer network. It is feared that a computer virus may have infected the network. Your security team makes the following statements:a. If
- The Dillies have five teenaged children, two boys named Ollie and Rollie, and three girls named Mellie, Nellie, and Pollie. Each is a different number of years old, from 13 to 17. There are three
- An advertisement for a restaurant at an exclusive club in Honolulu says, "Members and nonmembers only." Give two possible interpretations of this statement.
- The following newspaper headline was printed during a murder trial:"I am a liar" says murder defendant!Can the jury reach any conclusion from this statement?
- You meet two of the inhabitants of this country, Percival and Llewellyn. Percival says, "At least one of us is a liar." Is Percival a liar or a truth teller? What about Llewellyn? Explain your
- Traveling on, you meet Merlin and Meredith. Merlin says, "If I am a truth teller, then Meredith is a truth teller." Is Merlin a liar or a truth teller? What about Meredith? Explain your answer.In
- Next, you meet Rothwold and Grymlin. Rothwold says, "Either I am a liar or Grymlin is a truth teller." Is Rothwold a liar or a truth teller? What about Grymlin? Explain your answer.In Exercises
- Finally, you meet Gwendolyn and Merrilaine. Gwendolin says, "I am a liar but Merrilaine is not." Is Gwendolyn a liar or a truth teller? What about Merrilaine?In Exercises 61-64, you are traveling in

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