Problem 1E: LetP(x) denote the statement "x4. " What are these truth values? a)P(0) b)P(4) c)P(6) Problem 2E: LetP(x) be the statement "The wordxcontains the lettera." What are these truth values? a)P(orange)... Problem 3E: LetQ(x,y) denote the statement "xis the capital ofy." What are these truth values? a)Q(Denver,... Problem 4E: State the value ofxafter the statement ifP(x) thenx:=1is executed, whereP(x) is the statement “x1,"... Problem 5E: LetP(x) be the statement “xspends more than five hours every weekday in class,” where the domain... Problem 6E: LetN(x) be the statement “x has visited North Dakota,” where the domain consists of the students in... Problem 7E: Translate these statements into English, whereC(x) is “xis a comedian” andF(x) is “xis funny” and... Problem 8E: Translate these statements into English, whereR(x) is “xis a rabbit” andH(x) the domain consists of... Problem 9E: LetP(x) be the statement "xcan speak Russian" and letQ(x) be the statement knows “xknows the... Problem 10E: LetC(x) be the statement “xhas a cat,” letD(x) be the statement “xhas a dog,” letF(x) be the... Problem 11E: LetP(x) be the statement "x=x2." If the domain consists of the integers, what are these truth... Problem 12E: LetQ(x) be the statementx+12x.” If the domain consists of all integers, what are these truth values?... Problem 13E: Determine the truth value of each of these statements if the domain consists of all integers.... Problem 14E: Determine the truth value of each of these statements if the domain consists of all real numbers.... Problem 15E: Determine the truth value of each of these statements if the domain for all variables consists of... Problem 16E: Determine the truth value of each of these statements if the domain of each variable consists of all... Problem 17E: Suppose that the domain of the propositionalP(x) consists of the integers 0, 1, 2, 3, and 4. Write... Problem 18E: Suppose that the domain of the propositionalP(x) consists of the integers -2, -1, 0, 1, and 2. Write... Problem 19E: Suppose that the domain of the propositionalP(x) consists of the integers 1, 2, 3, 4, and 5. Express... Problem 20E: Suppose that the domain of the propositionalP(x) consists of the integers -5, -3, -1, 1, 3, and 5.... Problem 21E: For each of these statements find a domain for which the statement is true and a domain for which... Problem 22E: For each of these statements find a domain for which the statement is true and a domain for which... Problem 23E: Translate in two ways each of these statements into logical expressions using predicates,... Problem 24E: Translate in two ways using predicates, quantifiers, and logical connectives. First, let the domain... Problem 25E: Translate each of these statements into logical expressions using predicates, quantifiers, and... Problem 26E: Translate each of these statements into logical expressions in three different ways by varying the... Problem 27E: Translate each of these statements into logical expressions in three different ways by varying the... Problem 28E: Translate each of these statements into logical expressions using predicates, quantifiers, and... Problem 29E: Express each of these statements using logical operators, predicates and quantifiers. a) Some... Problem 30E: Suppose the domain of the propositional functionP(x,y) consists of pairsxandy, wherexis 1, 2, or 3.... Problem 31E: Suppose that the domain ofQ(x,y,z)consists of triples x, y, z, wherex=0,1, or 1, and z=0or 1. Write... Problem 32E: Express each of these statements using quantifiers. Then form the negation of the statement so that... Problem 33E: Express each of these statements using quantifiers. Then form the negation of the statement, so that... Problem 34E: Express the negation of these propositions using quantifiers, and then express the negation in... Problem 35E: Express the negation of each of these statements in terms of quantifiers without using the negation... Problem 36E: Express the negation of each of these statements in terms of quantifiers without using the negation... Problem 37E: Find a counter example, if possible, to these universally quantified statements, where the domain... Problem 38E: Find a counterexample, if possible, to these universally quantified, where the domain for all... Problem 39E: Express each of these statements using predicates and quantifiers. a) A passenger on an airline... Problem 40E: Exercises 40-44 deal the translation between system specification and logical expressions involving... Problem 41E: Exercises 40-44 deal the translation between system specification and logical expressions involving... Problem 42E: Exercises 40-44 deal the translation between system specification and logical expressions involving... Problem 43E: Exercises 40-44 deal the translation between system specification and logical expressions involving... Problem 44E: Exercises 40-44 deal the translation between system specification and logical expressions involving... Problem 45E: Determine whether SSx(P(x)Q(x))andxP(x)xQ(x)are logically equivalent. Justify your answer. Problem 46E: Determine whetherx(P(x)Q(x))andxP(x)xQ(x)are logically equivalent. Justify your answer. Problem 47E: Show thatx(P(x)Q(x))andxP(x)xQ(x)are logically equivalent. Problem 48E: Exercises 4851 establish rules fornull quantificationthat we can use when a quantified variable does... Problem 49E: Exercises 4851 establish rules fornull quantificationthat we can use when a quantified variable does... Problem 50E: Exercises 4851 establish rules fornull quantificationthat we can use when a quantified variable does... Problem 51E: Exercises 4851 establish rules fornull quantificationthat we can use when a quantified variable does... Problem 52E: Show thatxP(x)xQ(x)andx(P(x)Q(x))are not logically equivalent. Problem 53E: Show thatxP(x)xQ(x)andx(P(x)Q(x))are not logically equivalent. Problem 54E: As mentioned in the text, the notation!xP(x)denotes "There exists a unique x such thatP(x) is true."... Problem 55E: What are the truth values of these statements? a)!xP(x)xP(x) b)xP(x)!xP(x) c)!xP(x)xP(x) Problem 56E: Write out!xP(x), where the domain consists of the integers 1, 2, and 3, in terms of negations,... Problem 57E: Given the Prolog facts inExample 28, what would Prolog return given these queries? a) ? instructor... Problem 58E: Given the Prolog facts inExample 28, what would Prolog return when given these queries? a) ?... Problem 59E Problem 60E: Suppose that Prolog facts are used to define the predicates mother (M,Y) and Father (F,X), which... Problem 61E Problem 62E: Exercises 61-64 are based on questions found in the bookSymbolic Logicby Lewis Carroll. 62.... Problem 63E: Exercises 61-64 are based on questions found in the bookSymbolic Logicby Lewis Carroll. 63.... Problem 64E: Exercises 61-64 are based on questions found in the bookSymbolic Logicby Lewis Carroll. 64.... format_list_bulleted